Good point
I was not concentrating properly and was distracted by the fact that the registration numbers are the same on both barrett's car and mine (see below); but they are clearly not the same car!
My puzzle car is is very similar to woodinsight's; it's a pity we don't get to see his registration number then we might know better.
I think it's most probably the same car as woodinsight's and my information about it being Ausstin-based is maybe not right, so I'll merge my puzzle with that instead. It is possible that they made two, one with a Jowett engine and one with an Austin, as we do know they made a number of cars, but I think it's a bit unlikely they made two identical-looking ones.
Look here for barrett's car with the same registration number:
https://www.autopuzzles.com/forum/2017-46/barrett's-no-389/