I've not yet done this myself with the 1967 Big Healey I imported in 2001/2 but I udnerstand that the following is involved:
1.) Get the car in roadworthy condition and have it MOT'ed. Insure it too so that it can then be registered with the DVLA.
2.) You'll need to be able to document the car proving that you own it. To do so you'll need to have a proper title issued to show its ownership was transfered to you, or a bill of sale. Official documents like a US State title are better, I think- those prove that the vendor was officially recognized as its owner and you were a recognized buyer. This should show the vehicle identification number- the chassis number rather than the engine number. This should be shown on your Title or Bill of Sale. If it is not, it can be found (if you've good eyesight) stamped on the front suspension tower. See
https://www.healeyjournal.com/pages/geninfo.htm and click on the picture of the shock tower to see what's there. If all you have is a bill of sale, I'd suggest that you speak to the DVLA and see what their position is on that, I'd expect those cases would be handled individually on the merits of each, though I've no experience of that....
3.) You'll need to be able to show when and where it was imported into the UK, and that whatever import excise taxes and VAT were paid at the time, and have the Customs & Excise form (C & E 386) to show that. Note that the import duty on cars is 10% and then VAT is charged on top of that (and the hellacious thing is, VAT is charged ON the import duty, too- taxes on taxes!!!! made me nuts!) All this should have been done when it was imported and you should have that paperwork. If not...
I pre-paid the transport costs to the American shipper on mine to avoid VAT, I understand that now they will also consider these shipping costs for bring the car to the UK as taxable too.
It'd be best to read the Revenue & Excise
PAGE yourself. I phoned them and spoke to someone that was very helpful. I did that before I imported the car and was very glad I did.
At that time I was able to get a reduction because the car was considered a "Classic" and treated sympathetically. Now, I understand that they've stopped doing that because too many cars were being brought in. Now they're subject to the full tax, unless a very strong case can be made that the car is truly historic or unique in some way.
The DVLA will issue an age-appropriate registation plate for your car. Mine was built in 1967 so I'd get one with an "F" to denote that year, I'm not sure about the rest of it. Closer to the time when I register it I intend on asking to find out whether I'd get one reflecting a Glasgow registration where I now live, or if I could shop around and get something else from somewhere else. I'd like one with "USA XXX F" to show its origins.
In the past I believe that all imported cars were issued an non-dated "Q" plate, but this is not now done unless it's unclear when the car was made or first registered for road use. For most imports that should not be an issue.
My car is still a basket case, but I think I've all the necessary documentation for when I get it finished.
I'd suggest that you start getting it together well before you want to get the car on the road- if anything's missing it could take considerable time to sort out. More problematically, what do you do if they won't accept the documentation you have got? If you've bought a car someone else imported then chasing down all the paperwork if you haven't already got it may be an unwelcome challenge.