I assume you want shared hosting but do you have any preference as to the control panel, e.g. cPanel, how much disk space and bandwidth do you want etc.
Adding detail on your requirements especially anything out of the norm should result in people offering more appropriate suggestions.
A2B2 Hosting: Homepage offer shared plans, though I don't think they include backups. However I don't use their shared hosting for obvious reasons, but I do have a dedicated server with them which is used for gaming and so can say their network is good and their support is fairly responsive.
Before you sign with a host it is worth confirming that the server you will be on is actually located in the UK, some UK hosts in reality have their servers located in say Germany or the US especially at the budget end.