I am okay with BREN and No4 rifle in 303.
1937 is a bit late to switch horses imo
How
One option would be the adoption of the ZB53 in ground role (the AFV version became the BESA).
A rugged modern mmg - that is easier to build using modern practices and is lighter than the Vickers, uses a quick change heavy barrel rather than being water cooled and provided at 8 per battalion.
The Vickers would be rechambered to 7.92mm and use the same ZB53 metal link belt (I believe that it was the same used by the Germans in the MG34 and MG42) and continue to be used as a Division level weapon
The Vickers would be easy to rechamber and indeed modern private owners of these weapons often have conversion kits and can switch between calibres.
Ian McCullam aka gun Jesus (forgotten weapons guy) was selling his a few years back and was able to be converted to a number of calibres!
And ammo was delivered from corps/division in belts to the units so it does not matter really if it’s not in .303!
Obviously getting an MG34 would be great but the Germans went to great lengths to keep it and any documentation secret so it is unlikely that the British could get hold of it.
Again a parallel path.
Go for Garand over Mk4 ! Mk4 needs all new tooling, and most are going to be made outside the UK, Britain is too busy winding up for Bren.
An Enfield Garand with the changes BA suggested post WW1. Aperture sights (simple to train), 15rd mag over 5rd chargers (5s too hard to load in mud), heavier barrel and forget about the gas tap. Britain has Lewis and Vickers K gun with gas port!
Use the Japanese 6.5mm, converted to rimless 0.258" and fit it with a mark 7 type bullet. Less metal per round.
The Bren be made in .258", would have a straight magazine, interchangeable with rifle. Be lighter, and again less metal.
There is millions of .303" ready to be belted up for MGs. You cant throw that away. Only infantry needs SLRs in short term, the rest SMLE and Lewis.
Forget 7.92 and put energy into .258"