Moving physical server but keeping domain
Solution 1:
this is just a standard redelegation. general procedure is as follows:
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first, change the DNS on your current nameserver so that the web site points at the new UK-based host.
reduce the TTL for the domain at the same time. set it to 5 seconds or something low. make a note of the current TTL.
copy the zonefile(s) from your current name server to the new one, and set it up to serve your DNS.
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send a redelegation request to your domain registrar to redelegate your domain to your new nameserver.
at this point you'll have two name servers serving the same data, both pointing your web site to your UK host.
at some time in the future (at least as long as the original TTL before you changed it), you can increase the TTL back up to whatever it used to be.
even further in the future, at least 2 or 3 times as long as the original TTL, you can turn off your old nameserver. you only need to delay this step so long because there are a lot of broken nameservers out on the net that ignore TTLs and cache domain records for as long as they want.