My point is this - your website has an IP address, and there is a DNS entry existing that maps a .com address to that website IP address.
You can use DNS to map any other name(s) you like to that IP address, so you can get a UK-based domain registrar to register the .co.uk name for you, pointing its DNS entry to the existing website IP address. 1 IP address, 2 DNS names. Job done.
SD