BT Mobile services doesn't have its own cellular network as such. It was associated with the O2 network, which is not surprising considering that O2 was formerly called BT Cellnet and that BT retained a 50% stake in the company after the change of name. BT Mobile also had a deal recently with T-Mobile UK for some phones.
However, I believe that last year BT Mobile announced that it would be ditching both O2 and T-Mobile UK and putting everything over to the Vodafone network (a former "enemy"
). I can't confirm whether the change has actually taken place, but it was due to happen this year, so it might be worth asking which network you'd actually be under.
It's worth remembering that there are only five operating networks at present, so anyone offering service under some other name is using a "virtual" network under a deal with one or more of these five.
Hutchison is a newcomer setting up 3rd-geneation services and really has yet to become properly established outside of the major urban areas.
The four main networks are Vodafone, O2, Orange, and T-Mobile UK.
Vodafone has been around for years, and has an extensive network. O2, as already mentioned, was originally BT Cellnet.
The T-Mobile UK network was formerly One-2-One, which was part of Mercury Communications, the company set up in competition to BT after the telephone network was privatized in the early 1980s). Mercury later became part of Cable & Wireless.
The name changes, buy-outs, mergers, and so on can make it all pretty confusing. Of course, it's the same sort of thing as happened in the U.S. after the break up of Ma Bell, e.g. New York Telephone was split to become NYNEX, which was then taken over by Bell Atltantic, and is now Verizon.
The three-ring circus of the business side of the phone companies today would be worthy of any soap opera.
Edited to add:Oops, got so carried away on that tangent that I forgot to add this link which gives a very rough guide to the coverage for each network:
http://www.gsmworld.com/roaming/gsminfo/cou_gb.shtmlObviously the maps here aren't detailed enough to show all the local blackspots, but it's a start.
