When you say an MCA bill of sale, is this the one needed for Part I registration? If so, then you do need to contact the registry office in Cardiff to re-register or have the boat removed from the register I think.
If the boat is on Part I of the Register of Ships you should (but do not have to) contact the MCA in Cardiff 02920-448800 and re-register her in your name. They are usually very helpful if you call them. It's not compulsory but is a very good idea for various reasons, including proving your ownership, and getting a registration certificate to show to foreign Customs people when you go abroad. You'll have to fill in two forms and send this to the MCA with your bill of sale, plus a cheque for £124. You can also do this online, but as the rules and requirements are quite complicated it may be easier to talk to them first. The boats name must be unique - if any other registered boat or ship has the same name you can't use it.
If the boat is Part III registered (Small Ships Register) you can go online and re-register in your name via https://mcanet.mcga.gov.uk/ssr/ssr/
This gives you a registration certificate which is useful if you go abroad, but doesn't prove ownership. You can call the boat absolutely what you like - 'Queen Mary' is OK - so for the £25 fee you can get a certificate showing that a 6 ft rowing boat is registered as the 'Queen Mary'
Most brokers use the MCA bill of sale form even for boats that are not registered. It's simply a ready-made form. If you are never going to take the boat abroad there's no particular need to register. Just keep the bill of sale safely.
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....You'll have to fill in two forms and send this to the MCA with your bill of sale, plus a cheque for £124....
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You need to send Part 1 registration to the Registry of Shipping & Seamen in Cardiff. It is only £80 if the boat is already Part 1 registered and you are only registering a change of ownership. You also need to send the existing Cerificate of British Registry and a completed Declaration of Eligibility.
They are helpful on the phone.
It also states on the back of the Certificate that "A Certificate of Registry is not proof of ownership"