In the Post War there was a lot of effort put into small rounds and the guns to shoot them in both the west and the east. The efforts in the west were torpedoed buy the US with the 7.62 X 51 and since the US was paying for a lot of the weapons and ammo and would be the supply point for NATO incase the SHTF everybody caved into the US.
The US found that with newer powders they could get the same ballistics in a smaller case than the .30-06, but a slightly smaller rifle just means it is going to kick more
Then the US stabbed all their NATO allies in the back by unilaterally adopting the 5.56. Which then spawned a host of 5.56mm machine guns/squad automatics (belt and/or magazine feeds) most of which were not adopted.
Turned out the 5.56 could not do a lot of the things they wanted a machine gun to do. The little bullets could not carry enough tracer material to give a good trace out much past 300-400 meters. It wasn't as destructive to material at long ranges, It could not penetrate barriers as well. But this tell us very little about how well a roughly 6.5mm bullet of a little over twice the weight of the 5.56 would do vs a 7.5-8mm full power rifle load. Not as good as the full power round but a lot better than the 5.56 and perhaps 80-90% as effective for a smaller/lighter weapon was perfectly acceptable.
Some of the Military requirements were sometimes rather silly. The British .280 cartridge was turned down, in part but not just, because it would not go through both sides of a helmet at 1600meters(?). Now perhaps this had some sort of relevance in 1917 but in the late 40s or early 50s the days of 1600meter machine gun barrages were long over. Mortars and artillery and taken over from long range machine gun fire (which needed special sights, special mounts and special preparation like platforms, lots of sandbags, etc in addition to ammo and spare barrels).
But a lot of that isn't here or there, really. I was looking mostly at production and how far corners could be cut and still have a "reasonable" self loading or even select fire carbine. Main reasonable thing is that the gun won't blow up in the user's face, and provide reasonable combat performance. Not match accuracy, not great fit and finish, nothing like peace time or even modern levels of such items. But simple, no frills production and use, and decent performance, in this instance to or near "last ditch" levels.
And what production method would be best? Stampings and such were common for SMGs and would become common for assault rifles down the road (the StG 44 and StG 45 were heavily made out of stampings), and there were also castings and such. The Type 99 "Navy Special" last ditch Arisaka rifle did use a mostly cast iron receiver. But it was provided with a forged steel slug in the front of the receiver casting that formed a barrel extension and where the bolt lugs locked up. This, of course, is similar to an AR-10/AR-15 aluminum upper with a steel barrel extension or an AK or AR-18 or SA80 with their stamped receivers with a steel trunnion riveted or welded in place.
The US and British cheapened up some of their rifles a bit with stamped sheet metal barrel bands, trigger guards, cheaper sights and even going to two grove rifling at times.
British accepted a large variety of different wood for stocks.
But you are looking for even more savings.
Not all countries had the same ability to make things and some shops can stamp out a submachine gun receiver but not a machinegun receiver.
Using powerful cartridges and blow back means heavy guns. The less powerful you make the cartridge the lighter you can make the gun. Going to lock breech is more expensive but you can make the gun lighter.
Making cheap guns may make sense if the front line is only a few hundred kilometers from the factory. For the US where the front lines could be 10,000km form the factory building cheap, disposable guns may not work so well when you take shipping into account. Granted some soldiers could break anvils.
Different armies had different standards and even manual of arms. French bolt action rifles had no safeties. It was either loaded or not loaded. When they were importing them after 1968 the importers had to figure out how to add some sort of sheet metal lever that would block the trigger from firing the rifle, just enough to satisfy the government inspector.
This might never have passed a military inspector or design staff.
Dealing with rifle/carbine rounds that are several times more powerful than pistol ammo calls for different standards. The Short German 7.9mm and the Soviet 7.62 X 39mm generate about 4 times the power of a 9mm and they use a lot more powder to do it.
Some of this also needs looking at existing arms factories and equipment and what is needed for new production. Traditional arms factories were used to forging and machining.
If they stamped much of anything it was small parts (levers?) or magazines and/or the contracted it out. There were no large hydraulic presses in the factory.
In the US the M3 Grease guns were first made by a division of GM that made headlights. This was in pre sealed beam days so both the reflectors and outside shell were made from stampings. These guys were also used to making things for several different GM car makers and changing the styles (stamping dies) fairly often.
I do have my doubts about small shops/garages making submachine guns in quantity. I have no doubts about many shops making one or two parts under subcontract to be assembled someplace else. Again, different countries had rather different industrial capacity/experience. I am from the states in the US called New England, and that area had turned to manufacture in the mid 1800s and machines/machinery was rather common. Connecticut was home to more gun factories in the US than the rest of the country put together. It was also home to many companies making machine tools and with small companies going in and out of business, old semi-obsolete machinery tended to drift to small shops and barns/sheds. Some were for hobbies and some were used to make things as needed, many of them were used a lot more extensively during WW II.
My grandfather during WW II actually worked in a 4 man shop on the coast of Maine using machinery that ran on overhead belts to make parts for gyroscopes for Norden under subcontract. There were a bunch of small lathes, milling machines, shapers and the like spread out through the area/surrounding towns. But the ability to make stampings like the MP 44 used??? They had shoe factories and sardine canning plants. They could bend sheet metal in one direction at a time.
Other places in the country could do different things but mass produced stamped sheet metal guns needs a different 'plant/machinery' than normal gun making.
And it many need different plant/machinery than small metal stamping factories. In the 1970s I worked for several years for an alarm company and went into a lot of old Connecticut factories, some that made things like lipstick tubes back when they were metal and not plastic, Tableware (forks, spoons) and other 'stamped' goods. Maybe some of them could have been converted to make submachine guns, maybe some of them could not. You need both size and power for your stampings, maybe not car fender size but bigger than cosmetic tubes, razor blades or even lunch boxes.
With war time production you can order machines to do what you want and do it economically. This is different than peace time production where sometimes you have to figure in the cost of the tooling over the desired numbers of the production run. This also changed with time.
Stamped sheet metal can be a lot cheaper, but you have to be building large numbers of parts to pay for the machinery and the stamping dies.
It can be done but not if you are short of large industrial machinery to begin with.
And the more complicated the stamping the better engineers you need to figure out the changes in dimensions as the flat metal changes/stretches as it is bent. That or you need a lot of trial and error