This is never straightforward, you need to be prepared to dig around a bit and hope you get lucky along the way.
I assume you've got a VIN recorded somewhere, my first step would be to get a dating certificate (from the AOMCC, VMCC or Drags) for details of the factory spec and who it was supplied to.
Using the reg. no, also check to see if any of the old local authority registration records still exist anywhere. One of these links should help with finding which office allocated the original reg. no. https://www.drivearchive.co.uk/regcodes.htmhttp://www.cvpg.co.uk/REG.pdf
This site has details of which records survive from the various offices and where they are archived. http://kitheadtrust.org.uk/england-wales
(If you think there are any records you might have to arrange a visit in order to examine any archives in detail.)
I think the DVLA will provide a certain amount of data from the computer records - but only locations and dates of changes (or so they told me). I was advised to use form V888 but you might need to show that the bike is in your name - I'm still not completely clear about what they offer even after reading all the notes and talking to them over the phone....
Hope that's some use - best of luck!
'55 Huntmaster, '56 VH, ' 51 VH, '80 R100RT, '00 Sprint ST (but all those Ariel parts can only make one running bike...)