I have seen a photo of 1st Border Regiment in Palestine in 1938 wearing slouch hats, khaki, aertex shirts, with '08 pattern webbing shorts and rolled puttees. According to Mike Chappell this set a trend for slouch hats in Palestine.
It's another one of those transitional phases. On the whole though you would get away with Great War era figures in shorts and Wolseley helmet, Copplestone, Brigade etc. Pulp figures do Brits in collared shirts but they are all wearing '37 pattern webbing, which didn't really see widespread issue until just before the outbreak of war. All depends on how anal you want to be.
Worth remembering that Palestine isn't all glorious sunshine and balmy nights, it gets cold in winter and so figures in serge service dress are a possibility. Mix and match is another possibility. The same ref that has the picture of the Border Regt shows men of the Manchesters in Egypt in 1936 wearing service dress jackets and tropical shorts, topped off with tin hats.