I'm not a great fan of rat rods but that's not your question.
The body shape, original or not, will always determine the vehicle so a 1936 Morris stays as is regardless of the engineering beneath it.
In the early 80s we needed a transporter to take show worthy cars and bikes around the European shows. We converted a Ford A series chassis - shortened at the front lengthened at the rear - fitted with a Cargo diesel (from Ford's Jet Truck) and ZF box plus adding a third axle and made it into an artic. We handbuilt a two/thirds size Peterbilt body with sleeper. DVLA wanted photos with the engineer's report and then they changed the make to Peterbilt even though it had a Ford A series Vin number.
Replica cars - e.g. 1932 model B - get registered as Model Bs - assuming all the paperwork is done correctly.
Body shape recognition determines the model.
Regards,
Rodger