Well, it depends. With the mortar, clearances are a bit loose, length of the round is less important, etc. unless the 81 and the 82 have different firing mechanisms they likely would be nearly interchangeable.
The machine gun issue is quite a bit different. What we might call .51 cal is actually 13mm. This is a common metric caliber. Our .50 caliber dates back quite a while, like nearly a century (was designed by Moses Browning). But the way the two are chambered, the length of the round, the firing (rimfire/centerfire), the detailed shape of the round can all be different. No, you cannot just feed .50cal rounds into the 13mm gun.
Looking at Wikipedia, it looks like the 82mm and 81mm mortar difference dates back to before the Cold War but does seem like it was intentional.
The first 82mm Soviet mortar was the 82-PM-36 which debuted in 1936, and was based on the French Brandt mle 27/31 81mm mortar which itself influences a lot of other mortars fielded by countries like the United States, Germany, and Japan. Uniquely though every other mortar was a similar 81mm design while the Soviets were the only ones to convert the design to 82mm. And according to the Wiki page The M-36 could fire German 81 mm ammunition but range and accuracy suffered.
What .51 caliber machine gun? Soviet heavy machine guns were 14.5x114mm (.57 cal). Good luck trying to fire a 12.7x99mm cartridge with it.
No it wasn’t. Standard Soviet artillery calibers were 122mm and 152mm to NATOs 105mm and 155mm. Soviet tank guns were 100mm, 115mm and 125mm. NATOs were 105mm and 120mm. The 7.62x51mm NATO cartridge is not interchangeable with either the 7.62x39mm cartridge used by the AK-47 nor the 7.62x54mm which was originally developed for the Mosin-Nagant rifle.
It didn’t happen because it wasn’t designed to, but if you’ve captured large stockpiles of the enemies ammunition you’ve likely captured large numbers of their weapons as well. You’re more likely to use their captured weapons and find or create a source for more ammunition for these weapons to use. Look at Germany during WWII (the Marder series of self propelled anti-tank guns used captured Soviet 76.2mm guns for example) or Israel during the Arab-Israeli wars (Israel used large numbers of captured BTR-152, T-55 and T-62, often modified for their own use as with the Ti-67 and Ti-67s which replaced the 100mm with a 105mm gun, the Soviet engine with a US diesel engine, the Soviet 12.7mm DSHk with a US M2 12.7mm, etc).
Oh, and the DSHk (12.7x108mm) is presumably what you are calling a .51 caliber. Ammunition for the DSHk and the M2 (12.7x99mm) aren’t interchangeable either.
To further clarify, the Soviet/Russian caliber comparable to the US .50 is 12.7mm and it’s usually called that. As mentioned above the bullets of .50 BMG (Browning Machine Gun) 12.799 and Russian 12.7mm or 12.7108 are almost identical in diameter, the .50 BMG bullet is very slightly larger. But the Russian round’s cartridge case is significantly larger and it’s not practical for weapons of that power to run a smaller cartridge case through a bigger chamber.
The US military appellation ‘.51 caliber’ for the Russian DShk 12.7mm machine gun might have originated as a simple way to note it didn’t fire the same ammo as the .50 Browning without getting into the actual explanation of the difference.
13mm usually refers to the 13.2mm round used by the French, Japanese and other countries in WWII era. The case of that round (13.2*99 version) is similar in size to that of the .50 BMG but the bullet has a greater diameter. Versions of the Browning MG chambered for 13.2mm were used by both France and Japan alongside gas operated Hotchkiss types, the round generally being referred to as 13.2mm Hotchkiss. But again guns chambered for one couldn’t practically fire the other.
The US did something like that with a tank gun, too, calling one 76mm instad of 3-inch to make it clear that the ammo was different.
“It fired the same projectiles as the 3-inch (76 mm) M7 gun mounted on the 3in Gun Motor Carriage M10 tank destroyer and towed 3-inch Gun M5 anti-tank gun, but from a different cartridge case. The “76-mm” designation was chosen to help keep the supply of ammunition from being confused between the two cannon”
There is one case I am aware of. In 1942, British forces converted German projectiles for their own use.
British forces were using US M3 medium tanks armed with 75mm gun. Germans were using early Panzer IV also armed with 75mm gun. The US armor-piercing projectiles were rushed into production and had tendency to shatter when hitting armor. Some engineer had the idea to use the higer quality captured German projectiles. After some machining, he was able to fit them into US cartridges and fire them from the US gun. Apparently it worked so well, about 17000 projectiles were modified.
The case of German 75mm tank rounds* illustrates two issues which arise in using cannon shells in a weapon other than one for which they were designed:
it’s likely you have to have cartridge cases designed for the other weapon. As mentioned this conversion used German 75mm shells in standard US cartridge cases in US 75mm tank guns.
with a cannon shell there’s what is called a rotating band, a separate piece which engages rifling in the bore rather than the body of the shell doing it. These German 75mm rounds it happened could be adapted to the US 75mm bore by machining down the rotating bands. That wouldn’t be as practical on a machine gun bullet with no rotating band, the whole bullet would have to be machined down. Although it also would have been more complicated if trying to fire the US rounds from the German 75mm.
*probably the K. Gr. Rot 7,5 cm. The Anglo-Americans particularly liked the relatively large explosive filler in this round v early US APC M61 rounds which penetrated about as well and had space for explosive filler but none fitted yet, besides the M72 solid shot already on hand which had no space for explosive and which tended to break up as mentioned. The Germans decided the round was not strong enough when piercing heavy sloped armor and provided a much smaller explosive cavity in the Pzgr. 39 which some sources, apparently erroneously, assume was the round converted for British use.
Dismantling projectiles from cartridge cases is; but primarily was a routine job. The operations on fixed ammunition were done remotely by machine with explosive barricades/distance. The main hazard was ignition of propellant or primers; not the high explosive/chemical filler detonation/expulsion though that would occur after a slight delay in the event of a fire. Reloaders (hunters/target shooters) have small scale priming/depriming tools. Military and industrial companies have larger scale devices for artillery shells in the fixed or semi-fixed category. For images and explanation; see ammunition pull apart machines. ammunition pull apart machines - Google Search