Throughout this answer I will distinguish between the prerequisites to read and understand a text on its own, from the prerequisites to understand Christopher Tolkien's commentary (which will be comparing each text to other previously published versions of it, to indicate what was changed and when).
While it is debatable how much use there is in reading many of these early texts without being able to appreciate their context (e.g. it should definitely not be assumed that these are still in continuity with the published Lord of the Rings and/or Silmarillion), they can still generally be enjoyed as stories in their own right sans any of the commentary.
- The Book of Lost Tales parts 1 and 2
If you just want to cherry pick individual tales to read, you can probably get by with just having read The Silmarillion, and reading the tales out of order.
To understand Christopher's commentary you'll need to read these two volumes in order and to have also read the first age section of Unfinished Tales (or alternatively, the stand-alone Children of Hurin and Fall of Gondolin books.)
To just read the poems, it'll be enough to have The Silmarillion. To understand Christopher's commentary, you should have also read The Book of Lost Tales part 2 and either Unfinished Tales or The Children of Hurin.
- The Shaping on Middle-earth
The texts included in this volume have less interest on their own, but I suppose you again can probably understand most of them with just the Silmarillion. The commentary will depend very much on the previous three volumes.
- The Lost Road and Other Writings
There are essentially two separate books here. Part One is about "The Lost Road", can be understood with just The Silmarillion and perhaps The Lord of the Rings. Part Two continues the discussion from the previous HoMe books, and while again can mostly be understood with just The Silmarillion, will require all of the previous four HoMe volumes (and especially volume 4) to fully understand the commentary.
- The Return of the Shadow, The Treason of Isengard, and The War of the Ring
These three books discuss The History of the Lord of the Rings, and the only prerequisite here is The Lord of the Rings itself. To fully understand the commentary you'll need to read them in order, but even then only these volumes, not the previous five.
This book is also essentially two books.
Part One, "The End of the Third Age" falls into the previous History of LotR category. i.e. You can read parts of it (e.g. the cut LotR epilogue) having only read The Lord of the Rings, but you should have read all three previous volumes to fully understand the commentary.
Parts Two and Three are about "The Notion Club Papers". To understand these you should have read The Lord of the Rings, and to fully understand the commentary you should have also read The Lost Road section of volume 5.
- Morgoth's Ring and The War of the Jewels
These books show the later history of The Silmarillion, as well as other related late works about the first age. The actual content about the Quenta Silmarillion itself depends a lot on volume 5, but the rest of these books will be fine having read The Silmarillion. To fully understand the commentary these two volumes should be read in order, and the first five volumes (but especially volume five) should have been read as well.
- The Peoples of Middle-earth
This book is also essentially two books.
Part One, is the history of the Lord of the Rings prologue and appendices. What was said before also applies here. The Lord of the Rings itself is the only hard prerequisite, but volumes 6-9 should have been to fully understand the commentary. (And in this case the whole volume 9, not just the LotR part.)
Parts Two through Four are various late miscellaneous writings. They can be understood having just read The Lord of the Rings and The Silmarillion, but Unfinished Tales will help in some places.
- The Nature of Middle-earth
For this book, you'll need to have read the Silmarillion and The Lord of the Rings. This book is very light on the commentary compared to HoMe, but to appreciate the texts it would be highly recommended that you have also read "Myths Transformed" and "Athrabeth" from Morgoth's Ring, "Late Writings" from Peoples of Middle-earth, and various other parts of Unfinished Tales and the final three HoMe volumes.
