I have experience with both Sky and BT:
Had Sky at my parent's house - aside from the first two weeks after the connection went live, the connection was perfectly stable for the 4 or so years that I was using it. They still have the same line and the same router, and there are no problems with it whatsoever. It might drop the connection, say, once a month, if that...
I've been on BT where I'm living right now for about 15 months - my main gripe is with the slow speed I get, but that's a limit of my line and there's really not much I can do about that... In terms of the connection itself, it's generally stable, though it does seem to disconnect in peak time occasionally - usually either on a sunday afternoon or a Wednesday evening (might just be my local area?) - aside from those, when it's not disconnecting and taking 5 mins to sort itself out, it's generally quite good. It's as fast as my line can manage, and they did upgrade the upload speed not long after it went live too.
Not had to deal with BT tech support yet so can't speak for them myself....
If BT or Sky are the only two options you have, I personally would choose Sky, as I've had absolutely no problems with them whatsoever. BT are (in my experience) not that BAD, but they are slightly worse than my experience with Sky.