Do you have any more photos of the car, it is virtually identical to a Riley Sprite from the side.
No, unfortunately. The puzzle image was taken from a period advertisement, and there is always the chance that the garage sent the wrong copy to the printers. I am sadly aware of that. I must admit it is a bit dubious, and I have taken the ad as fact in good faith. Then there's the possibility that the builder married a Sprite body to the Bugatti chassis. The possibilities, whilst not endless, are there for us all to muse over.
I think I'll put a proviso on the puzzle heading.
PS I've just re-checked the ad. It was by Chiltern Cars, who always had an interesting and eclectic range of stock. The ad did not contain any other cars which were not pictured which would fit the bill of a Riley, if you follow my drift. I had wondered whether they had submitted the copy for the Bugatti-Mercury incorrectly attached to a photo of a Sprite, but it appears not.