I am not sure about registration in the BVI but assume that the conditions will probably be the same as in other British overseas territories whereby a vessel must be registered in accordance with part I of the UK Ship Register. This can be an exhausting task as the vessel's history must be traced from its first building and all transfer of ownership must be officially recorded. The vessel will also need to be surveyed and a carving note issued by a surveyor once the official number has been carved into the main beam (or similar).
The reason for all this rigmarole is that once a vessel is registered in Part I of he Registry its ownership is established and liens can be taken against that. Obviously this comes from merchant shipping and not yachting customs but so is it.
Part III of the register is solely for pleasure craft and grants a vessel British nationality but does not establish right of ownership. I think the fee is £25 and it is a really simple exercise which can be done on-line but:
1 - vessels can only be registered in the UK, not its overseas territories
2 - the person responsible for the vessel does not need to be British but even if (s)he is or not (s)he must reside in the UK. The issue of residency is however not checked but you will need address as the application will otherwise be rejected.
You will find that there are agents in, for example, Gibraltar, who specialise in vessel registration. Other possible alternatives to BVI may be Bermuda or Gib. In both cases the exercise is more expensive as an agent really will be needed.
If looking for other possibilities, I could suggest the Marshall Islands or Cook Islands, neither of which have flags which are likely to annoy others and both are relatively simple places to register vessels.
let me know if you need more help. Send a message or email through the forum, being perhaps a little more explicit re where you will be going and thus the reason for the change of flag and I will try to help.
Aye // Stephen