Hello there and welcome to the forum!
I've just done a move from one cPanel server to another and it was pretty easy. I would say tech support will be more than happy to help from the past experience I've had with them. The support here is great!
Regarding transferring your domains, I'm not sure whether eUK will transfer them in for free, however as you're current host seems so understanding, they may well be happy to just change the nameservers as and when is necessary to eUK. This would mean hostmonster would still control the domain name setting etc. but when you go to the
website hosting, you get your
website hosting on the eUK servers. I do something similar with my domain names as I've bought them from 123-reg and point them to the eUK servers. It's dead easy to do but they may have to do it for you (you'll need to ask them about that). If however it is free for you to transfer them to eUK, I would do that but you'll be best asking the billing department regarding that (or one of them might comment on here).
Anyway, hope that's some use! Feel free to shout for more help. That's what's great about this forum that someone usually knows the answer and no one cares if it seems like a stupid question. We've all been there!