This frame appears later by the fittings... and kind of plain.
It was my understanding he built in the USA for a long while, could have exported back to the UK, not sure.
What type of bottom bracket threading?
In the image of the bottom bracket adjustable cup, it looked like no threading indication. If Italian... hmmm.
The bottom bracket I think is a Silva (Italy) investment cast unit.
It would press the case better if other Cooper's with Columbus dropouts are around.
The fork is different
The stay ends are different.
The headlugs look a bit "plug and play" to me, typical to me would be more file work.
Other examples with the same rear bridge would push the cause too.

The internal top tube cable ports and the cable rings at the lower head lug are to me, modification later. So, I don't use them for assessment.