Hi- I'll offer some advice since I'm in the same boat (well, almost- I'm not engaged yet!)
I've been a PR of the US for about 13 years, and my boyfriend's British. After researching immigration options, I've found that the best route would be to get your US Citizenship ASAP because that makes things a whole lot easier. If you move to the UK, it gives you flexibility to come back and live in the US at some point (as well as not having hassles when you come to visit!), and if he wants to move to the US, you are immediately eligible for him to come over (and get a K1, etc.) You can marry him and then file to bring him over, but as a PR you are a lesser priority for US Immigration and you get I think it's 2nd or 3rd preference when it comes to the fixed allowed numbers of new immigrants each year (which does not apply to fiances/spouses of US Citizens) and so the application will most likely take a few years (as opposed to several months if you are a US Citizen.) Also, as a PR, you cannot file for a K1 to get him over to the US as a fiance and then get married- that visa is available for US Citizens only.
I haven't had any experience with the UK immigration system, but from things I've read/the hell I've experienced with US immigration, it seems to be a lot easier to move to the UK as a fiancee/spouse than the other way around! (Can anyone confirm..?) Hope this helps, and good luck!!