Name does not have to be unique or registered, infact few boat names are unique.
To go abroad you will need it registered (SSR) plenty of info on that on RIBnet found by searching. I once had a link to a site where you could search against boat name in a database of SSR registered vessels and was supprised how many names were duplicated.
To get a radio callsign (which is unique) and licence you need to register, which can be done online (the website excapes me, again search here or someone will post a link i am sure) and there are some names they don't allow - for example you can't call your boat "Dover Coastguard".
looks usefull, follow top ofcom link for online licence application