Pic. looks like late 1930's to me. Still wearing the old style jacket with stand and fall collar. I wore one of those in RAF blue in the late 50's (ATC). We did get issued with battle dress blouses and a magnificent fit and flare greatcoat though. Wish I still had it.
Have to wait for the military experts to see your Thread for a definitive answer. I did find lots of badges similar to yours, they were more associated with 'special forces'.