We have a UK king, bought to replace a US king (yes, I said US king) that was sagging after 3 years of use (and we are NOT big people and it was NOT a cheap mattress). I've never had an issue with fitting a UK king in a BR in our houses, but I do know some that would struggle. It's a bit tight in our current house, but doable, and works for us.
Personally, I would bring a frame. One of those metal frames on casters that you can buy practically anywhere would be cheap and cheerful for the short term, especially if you have a box spring - which I assume you do. If not, frames from Ikea will work. Bed linens are just that tad different and don't quite fit right is you get UK king for a US Queen. Plus, far better quality sheets for far less money in the US.
If you are shipping other stuff and you are happy with your mattress, bring it.