You can do a WHOIS search that will show you if a domain is registered.
If you want a .co.uk domain then click here and put the domain you want in the box on the bottom of the left hand side.
click here for an example of a taken domain
click here for an example of one that is still available.
For other TLDs like .com, .org etc you can use the same process but not with Nominet.
click here will be a good place to look for .com etc.
Someone like OneandOne (click here) will let you search for a domain and show you if the one you want is taken or not.