It'll be different in pretty much every case really.
When we went from TalkTalk to BT both companies were terrible. As soon as we called TalkTalk to end the contract, and discussed that they'd continue the service until a certain date, they cut us off that very day (about 2 months before the changeover). BT said they couldn't do anything because they didn't have control of the line until the agreed date, and talktalk said they couldn't do anything because they'd already put in the mac code which transfers the line...
If it's in a sky area however, I'd guess that would mean cable? So presumably your new service should be able to pick up right away? Or at least on the date of the agreed switchover? Although I'd still have thought a similar protocol would apply as if you did have a MAC code?
I just find it more hassle than it's worth to switch. Every company promises lower prices, better service etc at the start of the contract, but things always go down hill once they have you. Also, once you leave one company, I've noticed that they stop being any help to you whatsoever if you do have any problems during the switch. Half of them won't even give you someone to talk to unless you have a valid current customer number/account info - which you'll lose when you say you're leaving.
It's probably better to ask sky what's going to happen. I'd guess things would depend on your situation, so I'm not sure if anyone on here can say exactly how it's going to be.
What exactly don't you like about BT? I've not had any problems since the initial switchover.