Rather more British species of spider can give you a bite than is commonly realised, and these include some very common and familiar species.

In general, the cupboard spider is the most likely spider to give you a nasty nip in a domestic location in Britain
http://en.wikipedia.org/wiki/Steatoda_grossa
But in the SW, perhaps it is more likely to be the suitably named biting spider, a close relative accidentally introduced from the Canaries in the late 19th century, which has become widespread and common in the SW, and is spreading, and happy to live around our houses:
http://en.wikipedia.org/wiki/Steatoda_nobilis

Even common house spiders, small and large (Tegenaria spp), can bite you, as can the garden spider (Araneus diadematus), though they are unlikely to do so, and rarely very painful, though of course anyone can have a sensitivity or allergy.
http://en.wikipedia.org/wiki/Tegenaria_domestica
http://en.wikipedia.org/wiki/Tegenaria_gigantea
http://en.wikipedia.org/wiki/Garden_Spider

Out in the countryside, there's a Cheirachanthium with a nasty bite http://en.wikipedia.org/wiki/Cheiracanthium found in Britain, and the water spider Argyroneta aquatica http://en.wikipedia.org/wiki/Water_spider also bites.

In fact I rather suspect that most of larger spiders in this country can bite humans, even if they rarely do so, in addition to the noted smaller biters. Probably the beautiful wasp spider http://en.wikipedia.org/wiki/Argiope_bruennichi can bite, though that is not why it has its name. It's rare in this country and confined to the south, but getting more common. Certainly many of its relatives bite, including the silver Argiope which is well known as a biter in the Americas. http://en.wikipedia.org/wiki/Argiope_argentata