I think there have been similar threads to this before so it may be worth trying a search.
I just rang the Registry in Cardiff and asked them, they were very helpful but I know others on here got a very different response when they tried it. I must have got them on a good day
Oh, and don't try any of that "well we haven't actually bort it jsut yet cos final survey blah blah" cos evryone knows the survey will be hummy and hawy and you will buy it anyway unless it happens to be made of cardboard, and even then you will probly try a very low offer.
If you mean a searchable one on the internet, I do not think there is for the UK register (many others are though). I had a need to try not too long ago and had no luck - but the MCA site seemed to be under some sort of reorganisation, so maybe soon. I stand to be corrected however.
You may know this already, but every register I know of is searchable either by visiting (in which case there is often a charge) or by phone. If the Registrar of Ships is a friendly type, or the staff are, or it is their policy many will tell you on a phone call if your selected names are used or not - commonly they like you to give them three alternatives rather than sort of doing it one name at a time.
Don't know about the British Register, but if they do let you talk to them, and you find a desired name free, ask immediately if you can reserve the name pending your application. Many registers will do so for a limited period.
I had the same issue in Jan this year (for my March 2003 delivery ex yard).
Luckily I spoke to super MSA lady at the Earls Court show who advised that they were in process of producing a web site of names that were 'taken'.
After some sweet talk she opened her lap top and searched my 3 names for availability. Bingo - all were unused. So I applied the next day and got fixed up with my 1st choice.
The current system is nuts - you simply do not know what is available when you complete the paperwork.
Until 1980 one was able to buy a Lloyds Register of Yachts. This was a rather impressive green book with your name on the front in gold lettering. It was invaluable because it gave everything about a craft even the owners name. I have my own issues right back to the 1960s. Unfortunately (I assume) it became too expensive to produce and it fell by the wayside.
DeBretts Peerage produced 1 copy of their own but was never repeated.