I'm not an expert, but I've been researching visas for a year now. It's ALL crazy, I know...but thankfully everyone on this site is awesome.

SOoo....the K1 visa is only if you're settling in the US. That is the US's spousal visa. Are you planning on living in the States OR in the UK?
If you're planning on living in the UK, you don't need a visa to get married in the States. It's perfectly legal as long as you DO NOT settle here. Then, after getting all the supporting documents together (you and your fiance need to send in stuff), you would apply for a spousal visa. Whatever you do, do NOT purchase a plane ticket before your visa is approved. Lots of people make that mistake and lose money in changing flights, etc.
IF, by chance, you're planning on settling in the US, you will have to file for a fiance visa for your fiance. Then, once you two get married, you will have to file for the K1 visa (the spousal one). During the K1 process, your SO will not be allowed to work.
There are MANY different options you can go......it's crazy....lol. But worth it in the end.