Think you'll find that 02 (formerly Cellnet) are not connected (sic) in any way with BT.
As far as I can tell, 02 broadband are not an LLU operator and are therefore dependent on BT Wholesale.
The same goes for the non-cable version of NTL that others have mentioned.
Really your primary choice should be LLU DSL or BT based DSL.
If BT-based DSL, then the product portfolio of all ISPs can be considered identical, and it's the customer service that's the differentiator. I would second the name "Zen" already proposed above.
If LLU based, I've heard good things about UKOnline and Be.
To find out what non-BT based options you have in your area, use the search functions at SamKnows ....
SamKnows Broadband - Comprehensive Broadband Information