Working with machinist Harold Clisby, Jack Thomson constructed the first 'Thomson' prototype automobile in 1938. He designed and built his own engine, transmission, differential, braking system and uni-body chassis. Thomson used this prototype to secure investment from the Wiles brothers, with whom he then produced three Wiles Thomson prototypes under the company Wiles Manufacturing Company Ltd. The Wiles brothers pulled out of the venture in 1949. (Two of these earlier prototypes have been featured on AP.)
Thomson formed his own company, Small Cars Ltd, in 1950 and began work on what would be his fifth and final prototype automobile. Jack Thomson died in 1952, leaving the unfinished prototype to be completed by his son-in-law, Doug Giles. The puzzle car is that final prototype. It is not a 'Wiles Thomson', but the one and only 'Thomson' prototype automobile produced by Small Cars Ltd.
[Based on the image you've posted, it's possible Giles stuffed in the Ford 10 motor instead of the intended Thomson 2-cylinder and that he didn't complete the project until 1954. It's wrong to label the car a 'Wiles Thomson', however.]
The completed prototype and Thomson 2-cylinder motor:
