As Don Coyote says! I set up my own domains (one private, one Church, one flying group) many years ago, and have changed ISPs a few times since. Nobody needs to know I've changed, and there's no problem with e-mail addresses changing.
A friend of mine went with Sky, and wishes he hadn't. Their server will only let you send using the Sky e-mail address they give you, and it blocks stuff he doesn't want blocked. It's cheap, but it's user-unfriendly.
I use PlusNet in the Norfolk hideout, and it "just works". I get 7 - 8 meg pretty much any time of day or night - but this is a little village and there aren't many people to "contend" with.
My former neighbour in Essex is with TalkTalk, having moved her telephone to them, and she has terrible problems with it not working on various occasions. Once the contract period is up, I reckon she'll be going back to BT and PlusNet.
So: buy your own .uk domain - should cost about £6 for two years, with basic mail and web (no hosting). Set up mailboxes to forward to your ISP address, and webforwarding to your ISP webspace, and job done. When you change ISP, you go into the control panel and change a couple of settings, and job done.