Anyone exposed to it would ultimately die, leaving those in isolation the veritable heirs of the wasteland.
The Three Families would be dead, so only Mr House would be around if he still went on to seize and fortify the Strip. To what end, who knows?
The NCR would be long gone unless they controlled any bunkers they wished to use to safeguard important people, like military leaders or politicians.
The Enclave would be laughing, since they'd still have the oil rig and the Scorched Plague would not spread to them. However, they would still lose Navarro and would have to relocate them before TSP reached them. This also goes for any other currently unknown Enclave installations. They wouldn't be able to expand or continue their plans to reinstate the US Government.
The BoS would have to control their population and continue to generate water and food to sustain themselves. They wouldn't carry out their mission unless they threw themselves at a TSP infested wasteland in which they would probably die.
TSP victims eventually die of their affliction, meaning the only way after that to get infected would be exposure to the TSP directly. Advanced groups like the Enclave and BoS could develop technology, perhaps life support systems or filtration devices, to effectively survive in the wasteland.
Groups like the Shi, Enclave, Institute and BoS could, if it's possible, develop a cure for TSP, but they wouldn't be out of the woods yet. Scorchbeasts would run rampant across the wasteland, with their nests expanding and overpopulating, leading to more of them on the surface. If the Scorchbeasts willed it, they could hunt and kill the last of the humans and spread TSP through their hives, across the country. America would effectively be doomed, unless aforementioned groups had a way of killing them off and combating TSP.
Even with bunkers in mind, Scorchbeasts could technically break into them or collapse them by compromising the soil/rock/sediment etc. around them. Bunker-based factions would therefore have a vested interest in destroying them, and not just living out their days underground.
Vault 111 inhabitants would still be suspended in their cryo-pods. Vault 101 would remain undisturbed from Dad (James) and the Lone Wanderer in Fallout 3 and therefore wouldn't have opened. If Rivet City was established before TSP reached them, they could live in isolation. Same goes for Megaton, Tenpenny Tower etc. The Republic of Dave would perish and Canterbury Commons. Underworld, too. Vault City, Shady Sands and everywhere else unprotected would be finished. Nellis AFB might be able to hold out with their howitzers, however I'm not sure.
It goes without saying that Caesar's Legion would also be screwed, having a penchant for close combat and living outdoors.