It's often minor details that give the best clues, and when I saw this picture, I knew I'd seen the wheels before. The penny finally dropped today when I was looking for something else.
These are cast aluminium wheels used on some early Turner sports and racing cars from the 1950s. So pursuing that, I came across a picture, on the left below, of a Turner, thought to be chassis number 005, with the same registration number as the puzzle car, though looking quite different. However, the web page said that the car had been rebodied in 1956 in a style similar to that of the car known as chassis 004a. So I am guessing that the puzzle car is the rebodied version.
I can't find much detail about these early Turners, though this one is likely to have had a Vauxhall or Lea Francis engine. The pictures below show chassis 005 on the left, and for comparison, chassis 004a, which does look quite like the puzzle car.
It is thought this car was Chassis No. 005, Originally owned by Mr Ron Hill of Hills Fibreglass Developments Limited who would go on to supply Jack Turner with the first Turner 803 Body Shells. The car is not thought to survive today , However it is known that this car was rebodied around 1956 in a similar style to that of Chassis 004a. Registration Number 439 DRE is pictured here at Shelsley Walsh in June 1955.