a "bill of sale" would be a start.
if u dont intend leaving u.k.waters u wont need to register.
insurance not compulsery but u would be foolhardy not to have any
If its a small one, and you do not intend to go overseas, then paperwork is not required. Increasingly, harbours and marinas are insisiting on at least 3rd party Insurance, and it is foolish not to have that anyway as your boat can do a lot of damage without you ever going near it - going adrift, fire, sinking etc.
If you have a VHF transmitter you must have a Ships Licence, and it can only be used if a licensed operator is aboard and the paperwork for this must be kept aboard.
Many small boats change hands without any sort of paperwork following them, but if there is any question of crossing national boundaries then you will need to obtain documentation, starting with the Small Ships Register which at least gives the boat a unique identity number.
I have seen craft checked for a VHF licence at Port Edgar on the Forth and within the Caledonian Canel. Some time in the past, but have not heard of it recently.
In both cases they were more concerned that the craft had a VHF license if it spouted an aerial than whither there was an operators license.