Interesting idea, the question is, how much can Churchill do in this post without running afoul of the appeasement lobby (of which Baldwin himself was a member)?
I think an obvious problem would the conflict between the RAF and the Fleet Air Arm, and problems with engines for cruiser tanks.
A lot of procurement policies weren't finalised until 1938, most notably the Enfield .303 Mark IV and the Bren Gun. The Semiautomatic Pedersen .276 was being considered as an alternative for the former, while the Vickers-Berthier was still being produced as a Light Machine Gun. IOTL this led to a drum fed derivative, the Vickers K, would a belt fed version of this gun be feasible or even desirable? From what I've read the Vickers K was lighter than the Browning .30 cal but too heavy to be a true GPMG. There is also the question of SMGs, Evo Owen's design was initially turned down by the Australian government in 1938 with the British obviously more serious about rearmament, might he try his luck in the mother country?