Nautilus is one of the best engage CC tank in the game. Nautilus has so much CC in his kit and he is super simple to play as well. Nautilus has 4 different forms of CC in his kit.

His clear is decent thanks to the jungle pet.
You can full clear starting from blue or red buff in 3.21 with 1 smite (need practice).
I suggest to path top to bot 99% of time.

Pros

- Lots of CC in the kit
- Very simple kit to learn
- Strong ganks
- Hard to avoid his engage
- Can peel a lot in teamfights
- High burst with low counterplay with all the CC when ahead

Cons

- Falls off late game but not that much with this build
- If you get behind, you won't be that tanky
- Big hitbox, making it easy to get hit by skillshots
- Low dps so still need to rely on dps allies to win
- Easy kit but hard to play in the jungle as you need to hit your Q for ganks (I don't advise him for low elo and beginners) and clear is not that easy when learning the champion

From the wiki, here is Nautilus patch history :

Buffs :

V14.12
Titan's Wrath
New Effect: Now applies spell effects.

V13.3
Staggering Blow
Base damage increased to 14 − 116 (based on level) from 8 − 110 (based on level).
Titan's Wrath
Mana cost reduced to 60 from 80.
Base shield increased to 50 / 60 / 70 / 80 / 90 from 40 / 50 / 60 / 70 / 80.
Riptide
Base damage increased to 55 / 90 / 125 / 160 / 195 from 55 / 85 / 115 / 145 / 175.
AP ratio increased to 50% AP from 30% AP.

V12.22
Stats
Base mana regeneration increased to 8.65 from 8.63.

V12.10
Stats
Base health increased to 646 from 576.
Health growth increased to 100 from 86.
Armor growth increased to 4.95 from 3.75.
Magic resistance growth increased to 2.05 from 1.25.

V11.11
Riptide
Damage against monsters increased to 150% from 100%.

V9.8
Staggering Blow
Root duration increased to 0.75 − 1.5 (based on level) seconds from 0.5 − 1.5 (based on level).
Dredge Line
Base damage increased to 100 / 145 / 190 / 235 / 280 from 80 / 120 / 160 / 200 / 240.
AP ratio increased to 90% AP from 75% AP.

V9.6
Stats
Base armor increased to 39 from 35.46.
Staggering Blow
On target cooldown reduced to 6 seconds at all levels from 9 / 8 / 7 / 6 (based on level).

V8.16
Titan's Wrath
Base shield strength changed to 60 / 70 / 80 / 90 / 100 from 65 / 70 / 75 / 80 / 85.
Cooldown reduced to 12 seconds from 18.

V8.8
Stats
Base mana increased to 400 from 334.
Base attack damage increased to 61 from 57.54.
Dredge Line
Cooldown reduced to 14 / 13 / 12 / 11 / 10 seconds from 18 / 16 / 14 / 12 / 10.

V8.3
Stats
Base health regeneration increased to 8.5 from 8.374.

V7.22
Stats
New: Now has +15% bonus attack speed at level 1.
Base armor increased to 35 from 26.46.

V7.18
Dredge Line
Mana cost reduced to 60 at all ranks from 60 / 70 / 80 / 90 / 100.
New Effect: Now refunds 50% of mana spent when it hits terrain.

V7.15
Dredge Line
Base damage increased to 80 / 120 / 160 / 200 / 240 from 60 / 105 / 150 / 195 / 240.
Depth Charge
Cooldown reduced to 120 / 100 / 80 seconds from 140 / 110 / 80.

V6.16
Depth Charge
Knockup duration increased to 1 second from 0.5.
New Effect: Secondary targets are stunned for 1 / 1.5 / 2 seconds.

V5.22
Stats
Base mana increased to 334 from 284.

V5.16
Stats
Armor growth increased to 3.75 from 3.25.

V5.5
Titan's Wrath
Shield health ratio increased to 9 / 11 / 13 / 15 / 17% maximum health from 7 / 9.5 / 12 / 14.5 / 17%.
Riptide
Cooldown reduced to 7 / 6.5 / 6 / 5.5 / 5 seconds from 10 at all ranks.
Mana cost reduced to 40 / 50 / 60 / 70 / 80 from 60 / 70 / 80 / 90 / 100.

V4.17
Dredge Line
New Effect: Now pulls Nautilus towards champion-created terrain.

V4.11
Staggering Blow
Root on-target cooldown reduced to 9 / 8 / 7 / 6 (based on level) seconds from 9 at all levels.
New Effect: Ring indicator displays underneath already-affected targets.

V4.5
Stats
Base armor increased to 20 from 16.

V4.4
Titan's Wrath
Cooldown reduced to 18 seconds at all ranks from 22 / 21 / 20 / 19 / 18.
Shield health ratio increased to 15% bonus health from 10%.

V3.14
Staggering Blow
Root duration increased to 0.5 / 0.75 / 1 / 1.25 / 1.5 (based on level) seconds from 0.5 / 0.75 / 1 (based on level).

V3.5 (Balance update)
Stats
Base armor increased to 16 from 12.
Titan's Wrath
Cooldown reduced to 22 / 21 / 20 / 19 / 18 seconds from 26 / 24 / 22 / 20 / 18.

Nerfs :

V12.10
Titan's Wrath
Base shield reduced to 40 / 50 / 60 / 70 / 80 from 45 / 55 / 65 / 75 / 85.
Shield health ratio reduced to 8 / 9 / 10 / 11 / 12% maximum health from 9 / 10 / 11 / 12 / 13%.

V11.1
Stats
Base magic resistance reduced to 32 from 32.1.
Base health reduced to 576 from 576.48.

V10.16
Titan's Wrath
Base shield reduced to 45 / 55 / 65 / 75 / 85 from 60 / 70 / 80 / 90 / 100.

V10.2
Dredge Line
Base damage reduced to 70 / 115 / 160 / 205 / 250 from 80 / 130 / 180 / 230 / 280.

V9.22
Depth Charge
Base damage reduced to 150 / 275 / 400 from 200 / 325 / 450.

V9.12
Dredge Line
Base damage reduced to 80 / 130 / 180 / 230 / 280 from 100 / 145 / 190 / 235 / 280.

V8.16
Titan's Wrath
Base shield strength changed to 60 / 70 / 80 / 90 / 100 from 65 / 70 / 75 / 80 / 85.
Shield health ratio reduced to 9 / 10 / 11 / 12 / 13% maximum health from 9 / 11 / 13 / 15 / 17%.
Duration reduced to 6 seconds from 10.

V7.8
Stats
Mana regeneration growth reduced to 0.5 from 0.7.
Riptide
Mana cost increased to 50 / 60 / 70 / 80 / 90 from 40 / 50 / 60 / 70 / 80.

V6.5
Riptide
Base damage reduced to 55 / 85 / 115 / 145 / 175 from 60 / 95 / 130 / 165 / 200.
Maximum damage reduced to 110 / 170 / 230 / 290 / 350 from 120 / 190 / 260 / 330 / 400.

V5.22
Stats
Mana growth reduced to 47 from 50.

V5.13
Riptide
Base damage reduced to 60 / 95 / 130 / 165 / 200 from 60 / 100 / 140 / 180 / 220.
Maximum damage reduced to 90 / 142.5 / 195 / 247.5 / 300 from 90 / 150 / 210 / 270 / 330.

V5.5
Titan's Wrath
Base damage reduced to 30 / 40 / 50 / 60 / 70 from 40 / 55 / 70 / 85 / 100.
Riptide
AP ratio reduced to 30% AP from 50% AP
Slow duration reduced to 1.5 seconds from 2.

V5.1
Titan's Wrath
Base shield strength reduced to 65 / 70 / 75 / 80 / 85 from 100 / 150 / 200 / 250 / 300.
Shield health ratio changed to 7 / 9.5 / 12 / 14.5 / 17% of maximum health from +15% of bonus health.

V4.4
Titan's Wrath
Cooldown reduced to 18 seconds at all ranks from 22 / 21 / 20 / 19 / 18.

V3.9
Dredge Line
Removed: Traveling through some walls.

V3.8
Titan's Wrath
Base damage changed to 40 / 55 / 70 / 85 / 100 from 30 / 55 / 80 / 105 / 130.

V3.5 (Balance update)
Stats
Base armor increased to 16 from 12.
Titan's Wrath
Cooldown reduced to 22 / 21 / 20 / 19 / 18 seconds from 26 / 24 / 22 / 20 / 18.

Bug fixes :

V13.17
Fright Night Nautilus
Dredge Line
Bug Fix: No longer plays incorrect model animations when it was cast.

Unknown patch (V13.4 or V13.3)
Staggering Blow
Undocumented/Bug Fix: No longer locks his facing direction after the attack has landed, which would previously cause Dredge Line's Dredge Line's cast direction to be obscured if cast towards another target right after.

V11.15
Bug Fix: Recall sound effects now play correctly.

V11.11 - May 27th Hotfix
Titan's Wrath
Bug Fix: Fixed the AP ratio to its intended value (40%).
Riptide Riptide
Bug Fix: Fixed the slow to its intended value (30%).

V10.19
Depth Charge
Bug Fix: Can no longer stun Unstoppable icon displacement immune targets.

V10.13
Titan's Wrath
Bug Fix: Shield VFX no longer persists in the place where he enters Fog of War.

V9.16
Titan's Wrath
Bug Fix: Now properly triggers Electrocute Electrocute instead of granting only one stack while active.

V9.7
AstroNautilus
Bug Fix: Buffs aura now properly follow the model during certain animations.

V8.20
Dredge Line
Bug Fix: The channeling bar no longer displays weird text when casting Dredge Line.

AstroNautilus
Titan's Wrath Titan's Wrath
Bug Fix: Helmet VFX no longer linger after it has expired.

V7.23
Riptide
Bug Fix: The first ring now properly slows enemies hit.
AstroNautilus
Bug Fix: Shiny visor VFX no longer disappear when moving the camera away and back.

V7.16
Dredge Line
Bug Fix: Now properly stops if he collides with player-created terrain created in his path while pulling himself to his anchor.

V7.7
Depth Charge
Bug Fix: Fixed a bug where it would properly consume spell shields of champions in their path, but displace them anyway.

V4.12
Staggering Blow
Bug Fix: Cooldown indicator being visible to enemies.

V3.8
Titan's Wrath
Bug Fix: Damage over time AP ratio being lower than intended.

Aftershock is very good with this build since it gives some resistances that will scale well with all the Hp from the build and deals damage based on your bonus HP.

You should take shieldbash as it works with green pet to give you a second proc of shieldbash.

Conditioning gives "free resistances".

For the last Resolve rune, you can pick the 3 different runes based on your team and the enemy team.

Overgrowth is the standard rune because more tankyness.

Unflinching is good if the enemy has a lot of CC.

Passive : not much to say, switch target to apply it as much as possible to all the enemies. Try to chain CC and not apply all your CC at the same time.
You can use it to apply the 3 waves of your E (see below).

Q :
The cd is reduced when you hit walls, use it constantly in the jungle as you will have infinite mana for it. It's especially good for side lanes if it's pushed as you can Q, auto for the CC then E and you will have it soon after they flashed.
It can counter displacements like Blitzcrank, pyke Q, gragas R etc : if you Q a wall or an enemy just before the enemy displacement affects you, you will be pulled to it.
You can also use Q to force E 3 waves.

W :
It's an auto attack reset so use it just after your first auto attack.
The damage from it will last as long as you have a shield even if it's not from the skill. That's why green pet is good.

E :
You should try to hit 3 waves as often as possible especially early game and in the jungle since the damage will fall off later.

You can force 3 waves with one max range auto with passive then by going away from your target just 1/8s and pressing E.

You can also use a max range Q and press E the exact moment the anchor is touching your target. It's hard and require practice.

R :
There are a lot of different ways to use it depending on the situation.
Sometimes it's best to engage with it as you know your target won't be able to escape.
But most of the times if you engage with it, your target will just run away or flash or use a dash and it won't be really effective.
You should try to hit your Q and R later.
During a teamfight, you should always try to use R to a target behind the first enemy you hit with a Q(+E) + passive.

Combos :
The simplest combo is :
Q, auto(passive), W, E, auto

With ultimate you can do the same but with R at the end :
Q, auto(passive), W, E, auto, R, auto

With ultimate with less uptime CC but no counterplay (you don't want the enemy to flash or dash) :
Q, auto(passive), R, auto, W, E, auto

With ultimate but you optimise your E cooldown for more damage (need some ability haste, at least 30) :
Q, E during Q, auto(passive), W, auto, R, E, auto

His clear is now decent with pet buff and last season buff to his E.

You can full clear starting from blue or red in 3.21 with 1 smite. Rune shards are very important to increase first clear.
Here are the link with videos (patch 14.14):

blue start :
https://youtu.be/RruRClR7lcQ

red start :
https://youtu.be/8NlW1TCWAdU

I path top to bot 99% of time.

In the jungle, you get most of your damage outside of the jungle pet from your E.

This skill can actually hit 3 waves each dealing damage (2nd and 3rd dealing 50% damage).
Hiting 3 waves will allow max damage so it is very important to know how to do it and sadly it's not really easy.
You can learn how to do it in this video at 1.55 : https://www.youtube.com/watch?v=nBCPHQ6NWeI.
His E max damage against monsters is 165 / 270 / 375 / 480 / 585 (+ 150% AP).

With his W shield, he is really healthy during his first clear.

At 3 camps, you have the choice to continue farming with 2 points in E or to gank with Q.

With the recent changes to the jungle and the buff to his E, his clear is good enough to take 6 camps before scuttle spawn.

The most important to improve your clear speed as Nautilus is to learn how to double camp and not waste your E only for 1 camp if you can apply the damage (and the damage over time from the pet) to 2 camps at once.

Of course Nautilus is an awesome ganker so you should gank instead of farming but only if there is a good opportunity for it.

I prefer to clear top to bot because as a tank, the squishy bot lane is far easier to take down whereas some top laner can really snowball to the point you can't gank them.
Toplane is harder to gank in s14 also.

During the early game, you want to gank lanes that are pushed because Nautilus is not really good for dives unless the enemy is low.

You want to run to the enemy laner to auto and root them instead of engaging with your hook as much as possible.
You want to wait for the flash before to press Q.

If possible, I want to run at the target, press E then auto => W for the auto attack reset and wait for the flash.

If you have to engage with Dredge Line then do it, it's just not the best scenario.

At level 6, it is the same. Only engage with your ultimate if you are sure that if will be a kill, otherwise try to run at the target E, auto, W and wait for the flash then Q then your passive might be back again then press R if needed only.
It's really a devastating gank almost impossible to avoid for a lot of champions.

You should also use your hook on the walls if the enemy is close to a wall. It will reduce your Q cd by half and even at level 1 you will have it available to hit a champion during the gank. It's especially good on side lanes as the walls are close to the brushes and your enemies will most likely run through it.

I will continue to improve the guide over time but don't hesitate to leave a comment to give your feedback if you tried it.