Hello, I've been investigating various similar things as you, but in reverse (from UK, wanting to work in the US) and I'm married to a US Citizen
I'd just like to ask and bring up a few points. First off, is I assume you were some sort of soldier while a part of the military? You're not already a nurse/medical professional? And are looking to study to become a nurse in the future?
From what I understand about the military as well, is that your GI Bill is only for studying in the US, it won't be viable or transferable to any UK institutions. This means, you'll have to study and do your degree in the US, and will be waiting another 4 years at least before being able to move to the UK to work (unless you can afford out of your own pocket an English education, as I don't believe there's much support for international students)
Like others have mentioned as well, you'll need a student visa which is a very limited and specific visa. It might not allow you to work, and it probably won't allow you to stay past your studying time
Due to how nurses are trained in the UK as well, (again as others have said) you won't be able to practice nursing in the US either, as the NCLEX has certain requirements which aren't met within a UK nursing degree. You'd have to do some additional courses
Are you considering nursing as purely a route to living abroad? Or is living abroad just a great part of being a nurse? If it's the first, there are many other ways to consider working abroad, and there are other areas of skill shortages which might be more easier to obtain. I'd say do a quick search online for skill shortages in the UK, and see what they're offering work visas for
The best thing for you to do, I'd say, is get your education in the US with your GI Bill. From there, you can start looking and thinking about what to do. I keep hearing it's better to have a US degree if you want to work abroad than it is having a UK degree, as the US nursing degree is more "accepted". I know for example in Canada, they're adopting the NCLEX as their way of testing nurses and giving licenses, and I heard that US RN find it easier to get work in the UK than vice versa. This also means if you ever found you didn't like Europe and wanted to go back to your family in the states, you could still be a RN
I hope this helps, and good luck to you