If you decide to get married early to remain in the UK, you'll need to return to the US once you're married and apply for the spousal visa. Once you have this, you're able to live and work in the UK freely.
Alternatively, if you and your fiance have been living together at least 2 years, you may be eligible for the Unmarried Partner Visa, which gives you the same privileges as the spousal one, but would allow you to wait to be married.
Check out the Visa and Citizenship Board on this website for more detail on these processes.