hey, sorry for the very late answer, i hope i don't break any rules by digging up old unanswered questions.

To answer this question you have to understand how the adoption of a self loading rifle would have worked in britain in the 1930's.

To get a new rifle adopted, someone in the board ordnance has to be persuaded to the idea of mass issuing a self loading rifle to the rank and file of the British army. Already we are fighting an uphill battle for the following reasons.

A very limited amount of choice in rifles

halfway the 1930's there was a very limited amount of options for a self loading rifle. If we exclude all designs that aren't able to fire full power .303(the British standard munition of the time)or other full power rifle rounds we are left with weird enfield conversions and unproven prototypes(most of which they aren't even aware). The belgian design ( the later fn49) and french design(the later mas49) were still secret and incomplete at this time too. There is the pederson with a toggle lock, which causes problems in itself. And he zh29, familie of the zb-26(the gun the bren was designed on), but it also has problems that make it unsuited for mass adoption. The most realistic options would be the then avs36(later svt40) and the m1 Garand. Both those rifles would be patented so just copying them would not be an option. Lincensed production is very expensive and thus not really an option. Britain also still has a large arms production industry at this time, so they would design it themselves most likely.

.303 british

.303 is an obsolescent round at this time, it underperformed compared to other cartridges of it's time and was expected to be replaced in the 1910's with .276 enfield, but testing showed some problems with the munition and ww1 halted this change completely. The problem with .303 is that it is a rimmed cartridge , this causes issues when designing magazines and feed systems as there is a possibility to get the rims of the rounds caught on each other and causing so called 'rimlock'. Fully rimmed cartridges are fine for belt feds, but are very undesired for magazine fed solutions. There are ways to minimize rimlock, but no sane person would design a brand new mass issued main infantry rifle in .303(unless really desperate)

But changing to another caliber is not really an option at this time, the British empire has a massive stockpile of weapons and ammunition from ww1, and re-barreling existing vickers and enfields and producing all the new ammo is something that the british empire does not want to do in the economic depression of the 30's. They also just adopted the bren in.303 so .303 was there to stay.

Massive stockpiles of enfields

Like is said in my last paragraph, the British empire has an absolute massive stock pile of enfields of various types in their armories. Most of them were converted to the latest patern or were in the proces of being converted. Building a new self loading rifle and spare parts, and then producing enough for the whole armed forces, takes a lot of time and money, both of which britain didn't want to spend on this.

Doctrine and tradition

The British army had a well established infantry doctrine, and the enfield was an important part of it. Those guys loved their enfields, changing to a self loading rifle would require changing doctrine and military tradition, something generals at the time wouldn't have been keen on.

The British military command took years to remove the magazine cutoff from the enfield, a feature that was almost never used.But some people clinged to it for some odd reason, changing their mind about enfields would be even harder.

And we have to be honest here, a self loading rifle is better than a bolt-action, but the self loading rifle you would get at this point would not be a massive improvement. Add to this all the loss of experience with the platform and the gains would be fairly limited. Additionally they would have to rewrite manuals , retrain soldiers, change the logistics as they would undoubtedly use more ammo and so on. These things all take a lot of time.

So if we take all of this together, and we place our selfs in the shoes of the British person at the board of ordnance. We would have to do the following to adopt a selfloading rifle in the 30's :

Firstly we have to design it in Britain because there aren't any real viable designs on the market, and use another cartridge as .303. Which we do not want to do at this time, and if we designed it with .303, we would have to spend time and money preventing problems caused by the cartridge.

Then we have to pass military trials and probably refine the design, then probably do trials again(with the refined rifle). Then we have to convince even more generals and the rest of the board of ordnance that this self loading rifle is so much better than their tried and trusted enfields, and that their budget should not be spent on other things like more bren's, vickers, tanks, ... .

After all of that we have to refine the design for mass production(not always the case but prototypes are not mass production friendly most of the time.), and tool up one or several factories, creating jigs, tools, milling machines... . And the military command has to rewrite arms doctrine, and formulate a new manual of arms for the rifle, and so on.

After all of that we are probably in the same boat as france, russia, poland and belgium, they started development fairly early, hit some snags and before they could produce any meaningful number of rifles, ww2 had started.

Adopting a self loading rifle during the war is even worse, after dunkirk the british needed to rearm a very large part of their army. They needed to start cranking out rifles, submachine guns, bren's, tanks, ships,aircraft,..., by the millions. The massive undertaking of designing a new rifle and retooling factories at that time is all but impossible.

The USA was the only country to field a self loading rifle as their standard infantry rifle in ww2, they did not face many of the problems European nations faced for developing a new self loading rifle. And even then it took garand years to get his design right, and for production to ramp up.

As for the g41, that rifle entered the scene when the war had already started and had some major flaws in itself. Most notably using the bang system to trap gas, instead of a hole in the barrel. Therefor it was never issued in any large numbers.