I think with the GIL some people are overreacting: most python code is single threaded because of the GIL. So removing it doesn't actually break anything. The GIL was just making the use of threads kind of pointless. Removing it and making a lot of code thread safe benefits people who do want to use threads.
It's very simple. Either you did not care about performance anyway and nothing really changes for you. You'd need to add threading to your project to see any changes. Unless you do that, there's no practical reason to disable the GIL for you. Or to re-enable that once disabled becomes the default. If your python project doesn't spawn threads now, it won't matter to you either way. Your code won't have deadlocking threads because it has only 1 thread and there was never anything to do for the GIL anyway. For code like that compatibility issues would be fairly minimal.
If it does use threads, against most popular advise of that being quite pointless in python (because of the GIL), you might see some benefits and you might have to deal with some threading issues.
I don't see why a lot of packages would break. At best some of them would be not thread safe and it's probably a good idea to mark the ones that are thread safe as such in some way. Some nice package management challenge there. And probably you'd want to know which packages you can safely use.