If the hub is different, full width, half width then the angles the spoke holes are drilled may be different, I'm not sure if that's also the case with different size hubs, 5inch, 7inch for example?
First, all WM3 rime are the same width regardless of what bike it is on.
Now for you question, I have a 1929 AJS and looking at the parts list it shows some 350 and 500 machines having 20 inch WM3 rims and some 19 inch WM3 rims. It depends if your bike is a side valve, overhead valve or overhead cam. I don't have a 1930 parts list but I would assume much is the same?
As for your 1938 model 16 engine, I doubt it will fit the 1930 frame. AJS went out of business in 1931. After that other than the overhead cam bikes, all AJS's are a Matchless with different tank badges.
I think it qualifies too, but you should probably remind the forum members of its military heritage when you ask a question so it doesn't encourage non acceptable posting, what ever that actually means in a forum with no written rules.