Start by running a ping test:
http://pingtest.net/
You're seeking an 'A' or 'B' result from that test. If you get a lower grade there might be a fault with your phone line.
Then check that your router is plugged directly into a main phone socket (not into an extension). If your computer is connected wirelessly to your router, even though they're close to each other, switch to a wired connection. If you must use a wireless connection, try to keep your router and computer as close together as possible and away from other electrical equipment.
If you've not already removed the bell wire from your phone socket, do so (assuming that you've got the appropriate type of socket):
http://www.kitz.co.uk/adsl/socket.htm
(NB: Don't expect an immediate improvement. It can take several days, or even up to a fortnight, before BT's equipment recognises that your phone line can handle data at a greater rate).
However Tesco probably has a very poor 'contention ratio' for its 'Value' service. That's the number of other users who you effectively share an internet connection with at the exchange. Decent domestic ISPs have contention ratios of around 20 to 1. ISPs targeting their services mainly at small businesses might have a 7 to 1 ratio. (My own ISP has that ratio - but it's not cheap!). Many 'budget' ISPs, including Virgin Media and Tesco have a 50 to 1 ratio. However that reference to Tesco refers to their 'normal' service. If they're also offering a 'Value' product, the ratio might be 100 to 1, or even 200 to 1. So your download speeds are likely to drop dramatically at peak times.
Chris