if your fiance tells the border guards that he is coming to the US to be married, he very well might be denied entry on the grounds that they would automatically assume that he will try to stay illegally. US border guards just generally assume that anyone entering the US is going to stay illegally, and it is up to the visitor to prove otherwise. With a fiancee/spouse in the US, it gets rather difficult.
However, if your fiance brings proof that he will absolutely be returning to the UK on a specified date then he should have less difficulty, but there are no guarantees (I've known several SOs, boyfriends, girlfriends, fiance(e)s and spouses of USCs who have been denied entry, despite proof of return to home country).
Approved proof would be a letter from his employer stating that he is expected to return to his job on a specified date.
Copy of a rental/lease agreement to his residence, or copy of ownership of his residence (depending on circumstance)
Copy of proof of application, or proof of intent, for You to immigrate to the UK upon your marriage
I think on the USCIS website there is a list of approved documentation of proof of ties to home country. A K1 is not needed in this instance, and your fiance does not intend to remain in the US. However, not all border guards even know this! which can result in him being denied entry.
Anything that shows he is bound to the UK and must return is recommended. This isn't always necessary, but isn't always considered "absolute proof" either.
I'm a Canadian married to an American and just recently completed the whole green card process. I also know several UK citizens married to US citizens who are currently going thru this process, or have completed the process. In some cases, in either and both directions!
My husband and I are now considering emmigrating to the UK.