Part 1 was created for ships, as and when private pleasure vessels appeared they basically had to use this Register if going abroad (albeit the requirements to Part 1 register small vessels and non commercial are different - yer 30 foot yacht does not have to meet the same requirements as a Supertanker

).
However their was (and still is) no requirement for a UK boat to be registered in the UK.
Back in the mid 80's (I think) the French (& the EU?), but definately the French

said that they wanted all foreign vessels visiting their waters to be registered. No doubt a numbers thing as historically UK boats did visit French waters with no registration papers and pas de problem.
The UK Govt realising that the existing Part 1 registration would be too cumbersome to administer / looked for an easy way around the French problem - and came up with the Part 3 register aka the Small Ships Register (SSR) where the requirements to register were basically nil and on demand. I have not tested it, but apparently you could get a Lawn Mower registered on the SSR
Part 2 is a bit of a mystery to me

Possibly something to do with commercial fishing vessels if memory serves me correct

. or not
I haven't looked at the SSR requirements for a while, I recall that it is solely for British Nationals (plus a few others from the Colonies?) but apart from having a UK address (to send your documents to) I don't know whether they actually check up on this - of course this means that your registration would not be valid. Which may or may not be a problem depending on circumstances.......and BTW the SSR is
not proof of Title / Ownership. SSR works all around the world. It might be (arguably?!) a sh#tty register but is an official UK ship register so therefore has to be accepted by other countries.
A big difference between SSR and Part 1 is that SSR cannot be a company registration. The usual 100% legal work around for non UK nationals to have a UK registered boat is to Part 1 register via a UK company. Of course Part 1 registration is not without cost (I forget the exact figures all in - but vaguely I think £500 - £1000) and then maintaining a UK company via proffesionals is not cheap, but if able to DIY (with a UK address from Family / Freinds) not outrageous on annual cost - under £100 - with a bit of PITA form filling each year.....and BTW Part 1 is also
not proof of Title / Ownership (the certificate actually states that) although many believe that it is - albeit in practice it pretty much is, just not 100% (in fact nothing is 100% conclusive on ownership with UK boats). Note also that although mortgages can be reqistered on the Part 1 register their is no requirement for a lender to do so - although I am led to believe as the capacity to register a loan / mortgage is available but not used that not doing so may affect the lenders security / chances of recoverability - but I would not want to test that one out personally

Notwithstanding the above Part 1 is preferable to have (inclduing for resale so a buyer can easily get a loan secured) so although I would not bother with a Part 1 reg from scratch I would not let one lapse.
Of course an important thing to bear in mind is that where the boat is registered has zero to do with her VAT status! Therefore no reason why you cannot change her Reg to the USA - she will not automatically lose her VAT paid status for that. Obviously hanging the stars and stripes off her stern will make her look a little bit more non EU than a Red Ensign

and therefore more likely to be non VAT in the eyes of an EU customs officer - but your passport will do pretty much the same anyway......the important thing on the VAT is to have the documentary evidence that the VAT has been paid.
Some UK folk with UK boats can be very relaxed on the VAT documentation side of things - and opinions do vary accross the web

- but I would never buy a UK boat without being able to prove she had been VAT paid or I didn't want to use her for extended periods in the EU or she was 15% cheaper........on the basis that at the moment no proof of VAT paid is not my problem, after I bought it could be - and why would I want to take that / any risk when other boats with all the paperwork are available?.....but, as I said, opinions do vary accross the web on boat VAT

Folk with VAT paid boats but
no docs can sell them to anyone they want. except me