After having lived here over 6 years (and now actually being qualified for citizenship), I've managed some contacts with other Americans here and there. You can visit several ex-pat forums for Americans and get a pretty good idea. From my experience, many of us come over and get citizenship based on the following in a slightly descending order:
1. Ancestry. Quite a few of us have some close enough relatives to make this work.
2. Students who then get married to UK citizens.
3. Transferring through a company, and for whatever reason, deciding to stay (my favorite!
).
This is only my own feel from visiting ex-pat forums and meeting fellow Americans here.
As the previous poster mentioned, being an American doesn't generally give you any particular advantage, unless ofcourse you qualify for the ancestry option, which is perhaps a uniquely higher possibility for us. Certainly having English as a mother tongue and having American education credentials does help in several regards, but otherwise, you will have to hop through the same hoops as anyone else.
That said, do I think being an American helped me? I kinda do. My wife is Asian Indian, so I've gotten to know plenty of people in that community as well. It certainly
seems like I had a lot less scrutiny over my details every time I've had to get a visa or an extension. Things that took others months seemed to only take days or weeks for me. But this is purely anecdotal. There are certainly instances where non-Americans I know have come over as relatively easily as I did.
Good luck, and make sure to check out those ex-pat forums. They are a great resource for first hand experiences from an American point of view. For immigration advice into the UK, this forum is the best on the net.