Your options are one of the BT IPStream ISPs (the majority) or an LLU (Local Loop Unbundled) ISP. These are ones who have their own equipment at the end of your line instead of BT equipment. Depending on who the LLU supplier is, they may or may not offer phone services as well, including the line rental. For example TalkTalk have excellent deals (on the face of it).
The LLU supplier of choice is Be - though beginning to have problems caused by their own success - too many customers.
It appears your only LLU supplier is Talk Talk. They have a lousy reputation, but I think again that was due to huge take-up when they came to market with their broadband services. (My brother - internet-ignorant, broadband-ignorant, dial-up-user has just signed up to them on International 3 - I cannot fault his logic as he has 3 friends locally who told him they had intitial problems but now it is good and his wife phones relatives in Australia 1-2 times a week. His Talk Talk cost is less than his BT phone bill and he is going to get broadband).
Check with Be - my availability info may be out of date.
If not LLU, then of the ISPs selling BT IPStream services - as I said, the majority - then your best bet is one of the Enta resellers, the service with all is identical, the price varies slightly and the add-ons are different. Popular ones are UKFSN, vivaciti, ADSL24. Falconnet and Aquiss are others I know of - there is a site listing them somewhere. You do not get unlimited, but probably more than you actually want, as long as you organise yourself a bit.
Others to look at are IDNet and Zen. Maybe others but that's my view at the moment.