I telephoned the MCA, asked for the relevant dept., and asked them. I wanted to change my boat's name for part 1 registration, and found that I could find a version that was exclusive to me. Then all I had to do was pay up. At least I think it was the MCA. But it was where you send your aplication. They were certainly very helpful.
Its a good place to start, but as far as I understand it only logs against radio licenses and so may not capture all Part 1 names. (and may capture some names not on Part 1)
As far as I know, the whole Part 1 database is not online, and the agency usually ask you for a couple of alternatives. Many people try the 'of wherever' trick, to get a unique name, but if you dont want to do that, and only enter one name on the form, the agency will bounce the request. Just change it and resubmit.