Built up car form.

Gonna need a lot more info than that to help I'm afraid. What's the history of the car, is there any proof it's been registered before (here or abroad), has it been substantially modified, is there a possibility of getting dating certificate for it if it's a well known marque, and so on.

The process for registration will vary depending on what it's history is and what you can prove. All these routes will require different forms and supporting documents, and can have different outcomes.

Old American car, has a UK plate on it. Totally standard but partly dismantled, original vin plate is clearly visible. Owner has applied for v5 but told it needs the built up form as it has been off the road since mid 80s.
Not a big price for it, needs a bit of welding, but the big gamble is if the DVLA will play or not.

If its standardish then there's not a problem if the ID is ok, I would personally ring the DVLA and find out exactly what they think is happening with it, I'm guessing they've got it down as a hot rod of some sort?
Is there an option on a built up vehicle form to say it's all original or by filling one in are you admitting in their eyes that you have a vehicle with questionable history?

As Dez mentioned, you could treat it as a "new" import, although they may have some record of it after previous dealings...

On that note, the V55/5 and evidence you need to register an import is one of the forms you need to re-register a pre-1983 taxed vehicle (covered in my link above), so you'll need to do most of the same legwork/paperwork anyway.

For a re-registration, you then add to the paperwork a V765 (confirmed by a club) which just allows you to also (probably) keep the registration number.

In some cases this may not really be worth it, as on a simple import route, you'll get an age related plate anyway (assuming on either route you can prove dating), just not the one it has now. On a dateless pre-65 plate maybe you'd want to fight for the original, on later ones maybe not so much. You wouldn't be allowed to transfer out a dateless plate anyway so there's no financial gain.

The new plates they issue are unused older numbers, so to a trained eye you can spot it's not original, but most won't know or care...

Edit: have you run it through the online tax/MOT checker? If it comes up as not found, it's probably last taxed pre-83, so DVLA will have very little if anything in their computerised records, which is what I've assumed. Other possibilities are scrapped, exported etc. so you should check with DVLA what exactly they do and do not know.