My father was one of the doyens of A7 racing in the U.K, and having in my younger days had considerable experience and ownership of Austin Sevens, and still finding odd bits of them in my toolboxes and so on, I feel qualified to answer.
To begin with, a 1935 Austin Seven racer is almost certainly a special built on an A7 saloon chassis. There were works cars in 1935, but they are as rare as hen's teeth and all accounted for.
There was a rash of special building in the post-WWII era in the U.K, and the A7 was a popular and plentiful choice of car as foundation. The first 4 Lotus were built by Colin Chapman on modified A7 chassis, though sometimes with Ford 10 engines. The start of 750 formula racing encouraged many to use this as a start to motor sport.
As to how much it's worth - it all depends. A look at the picture would help!
Aeronca - let me correct some delusions. Real A7 racers are not all Ulsters. The attachment is of me driving an Ulster back in 1973. There were several other genuine A7 racers, including single-seaters. Ulster replicas are recent fakes, to put it bluntly.
Home-brewed cars are not called "Brooklands" unless you're seeking to deceive. There were genuine Brooklands cars, built by Gordon England, but they were much earlier. The white car you illustrate is NOT a Brooklands by any stretch of anyone's imagination, having been built long after Brooklands closed! I've seen that car before, but do not remember its constructor. It's a Special.
Your chassis, apart from being a basic A-frame, appears to have nothing A7 about it at all.