I assume you're working on the principle that the British continued with the pre-WW2 idea of changing over to rimless ammunition?
Of course the move to the 7.92x57mm Mauser ammunition would have meant that the Germans would have had a lot of ready to use ammunition available to them after Dunkerque (Dunkirk).
Still, I like the idea of the Brit's taking on some more Czech equipment - would make for some interesting early-war battles.
Even more interesting if the process had begun earlier, with the LT vz. 34 or LT vz. 35, & had led to a proper treaty between Britain & Czechoslovakia (rather than via mutual treaties with France).
Prior to the end of the war the UK had decided to standardise on 7.92x57mm Mauser (this may actually have been pre-war), it was only the evaluation of new intermediate cartridges that resulted in .280, EM-2 an Talden Gun that prevented this from happening.
As for the rest of it:
-the RAC did evaluate the LT vz. 38 but rejected it apparently because vibration made accurate shooting on the move impractical / impossible. Without a doubt this vehicle would have been superior to the various Vickers Light Tanks in service at the time and more durable than the early cruisers (the caveat being so long as Nuffield's were kept away from them so they couldn't do stupid things like deliver the tanks with not lube oil, including driving them to the trains, then to the ships)
-the BREN was evolved from the ZB vz.26 which was chambers for 7.92x57mm Mauser.
-the BESA, an evolved ZB-53/TK vz.37, retained 7.92x57mm because pre-war it had been desired to move to a rimless cartridge but time ran out, it was feared that the industrial, technical and logistic difficulty of converting the gun to .303 would be worse than retaining the original calibre (RAC had their own logistics). It was also thought that it would be beneficial to be able to use captured German ammunition.
-in addition to their pre war interest in rimless cartridges the UK was also investigating semi automatic rifles but ran out of time. I do not know it the ZH-29 was ever assessed by the UK but interestingly Vickers Armstrong did licence produce the Pedersen .276" Rifle.
-Hobart was a leader in armoured warfare strategy and tactics, with Heinz Guderian following his work (even apparently paying for it to be translated). He raised and trained the Mobile Force (or farce as dismissed by some senior officers) which grew into the Armoured Division and then was renamed the 7th Armoured Division, he also raised and trained the 11th and 79th armoured divisions before converting the 79th into a specialised armoured engineering and assault division.
-Hobart was forcibly retired in 1940, becoming a Lance Corporal in the Home Guard, before being reinstated in 1941 at the direction of Churchill. Other attempts were made to sideline him by the establishment on health grounds.
-the 79th apart from its amphibious tanks and various armoured engineering platforms was also the command formation for both the 1st Canadian Armoured Personnel Carrier Regiment and the 49th Armoured Personnel Carrier Regiment.
Basically I have put a lot of different pieces together, something that only could have happened if Hobart had been held in higher regard early in the war. That is probably the real what if, i.e. Percy Hobart was given greater authority and support pre and early war.