I compared oil pumps between a 1936 Aero 600, a 1935 Cotswold (350cc) and my A31.

First thing I realized is that it seems my A31 had the oil pump replaced at some point with that from a big twin! However all is not lost as they are somewhat interchangeable. In the next image the arrow shows where a notch has been sawn out of the oil pump on the A31.

This is done so it fits within the smaller perimeter of the 250/350cc engine. The pump fitted to the 1935 Cotswold would appear to be what the A31 ought to have. The difference between the big twins and the lightweights is they used two pegs to locate the big twin oil pump, but did not bother for the lightweights. The location of these is shown in the next image:

The mounting bolt pattern, oil galleries cast into the pump face, and ports in the timing cover are identical between these three engines.

The gear and worm are also the same. One other difference, though not impacting you, is the oil spray pipe is longer and has a jog for the 500/600cc compared to the 250/350cc.

In conclusion I would say the oil pump from the A31/A32, 1934 Y/Y1/Y2, 1935 Comet/Cotswold, or 1926-37 Aero 250/350 would be the same. You could also use the oil pump form a 1936-38 Aero 500/600cc, but would have to eliminate the locating pegs and notch the base slightly to clear the timing chest joint face.
-Doug