A good, cheap custom domain site for my blog?

[Open Sauce]

Firstly....I'm also from the UK....it does not matter where you buy your domain name or web hosting from, so you can purchase either from a US company.
Now....your question is a little confusing because it's not clear if you wish to only purchase a domain name which will point to your tumblr site, or if you wish to create your own site to replace your tumblr one.
If you wish to keep your tumblr site but have your own domain name, then you can purchase a name from as little as 2 pounds from crazy domains: http://www.crazydomains.co.uk/
Once you've purchased the name you will then have to point it to your site at tumblr. To do this you will first need to find out what nameservers to use for tumblr, and their website should tell you.
Once you know thhese, you will need to log in to your account where you bought the domain name from and choose the option to set your nameservers. These will be something like ns1.tumblr.com and ns2.tumblr.com.
Then.....wait for between 2 and 48 hours for the name to propogate the www.
If you wish to start your own blog separate to tumblr, then you need to purchase both a domain name and web hosting, and if you buy them at the same time from the same company, they will set the nameservers for you.
Once your name has propogated the world wide web, you will then need to set up your blog, you could either download wordpress or something similar by doing a google search, then upload it to your site and install it, but it would be much easier to purchase hosting which includes an auto installer. Try hostgator for a good deal: http://secure.hostgator.com/~affiliat/cg...?id=adbusy