I don't actually see this as a "problem" for 2 reasons. 1) SS2 includes the door disabler holotape option, which allows you to access Depravity's version of the interior cell. You can use that toggle to choose which interior version you want to access. And 2) Once you've entered Depravity's version of Fallon's, you no longer need to use the exterior door unless you play in Survival mode, because you can directly FT into Depravity's interior cell.
The destruction of GNN and the added residual radiation is the number one conflict between Depravity and SS2 Ch2. But I am currently working on a "patch" that will allow both mods to work together and avoid that conflict. But I'm not sure which conflict you're talking about in regards to the Nightingales and the hospital, I think I might have an idea which quest you're talking about though. So yeah, as long as people using both mods are aware of a possible conflict, and know to complete Depravity's quests first, it should be good? I haven't started Ch2 yet, so I'm not personally aware of what the conflict would be.
As noted above, I am currently working on a patch that will cease the damage to the GNN building in Depravity while still allowing that mod to progress fully and with minimal impact to its storyline. The door barriers are only one part of the problem, and easily removed without the CK by just selecting them in console and disable/markfordelete. The bigger problem is the radiation, but I'm getting that taken care of.
I was able to run Depravity and the other Thuggyverse mods with Ch1 without any problems. The only issue I ever had with The Ron was way late in my game, after I found pretty much every SS2 unique settler, I went back and asked if he knew of anyone wanting to find a home. He gave me an objective to find someone, but a marker never showed up. Using the SQT/SQO/SQS commands on his quest, it states the door/path to his target is unreachable. I went through every DLC location trying to find the marker, nothing. Depravity changes Hotel Rexford and PV changes the institute, but both original interior cells are accessible by either terminal or holotape. And after checking, yep. New unique character Benni was inside the original Hotel Rexford upstairs. Because that interior cell's location is only accessible via terminal, the quest from The Ron wasn't able to find an accessible door. But the problem is now resolved.
The only problem with trying to use both mods and simply not blowing up GNN is that you will halt any way to progress further in Depravity's quests. This means you won't have access to the DC management, won't be able to take over the New Rexford, won't be able to hire a few unique characters as employees, and will be unable to complete some of the vanilla DC quests that only become available after you take over DC as temp mayor. That's actually quite a bit of content you'll be missing out on.
But the big news is that, thanks to Playerlin on Thuggy's discord, I have a huge starting point to creating a patch that will allow both mods to be used without the GNN problems, which is the main incompatibility issue between the two mods. I know a lot of people have been asking for a similar patch; on these forums, Steam forums, reddit, etc. I think I have most of the data needed to get the patch done, but I lack the CK skills and FO4edit wizardry to make an actual patch. If anyone else with knowledge/experience doing patches or small mods is able to assist, I'd be extremely grateful for the assist (and I'm sure a lot of the community would be as well).