Some reading:
The photo of AMP366 comes from here
https://motorcycletimeline.com/1935-2/ where it was captioned: "Three years after acquiring its first BSA 498cc V-twin for assessment the War Office ordered a lightweight version (hence the undersized fuel tank) but subsequently decided a side-valve single would be more suitable."
Here's a similar model that was for sale (some time ago)
https://motorcycles-for-sale.biz/sale.php?id=37628 The model was called the B15 - some interesting comments on "standardisation" that happened on the military models. The tank on the bike in the photo with your grandad is most likely BSA C11; perhaps part of the standardisation process.
Leon