It sounds like you're misunderstanding the roles of the the two guns. The MG42 was a rifle-caliber general purpose machinegun (GPMG), while the M2 Browning was a heavy machinegun (HMG). Whereas a GPMG is intended for flexibility in roles as both a light machinegun and mounted machinegun (on tripods or vehicles), HMGs - especially large caliber HMGs like the M2 - are designed purely for fixed mounts. Heavy caliber guns like the M2 are also employed more as anti-vehicle weapons than anti-infantry weapons.

But getting to the core of your question - some tried, and ultimately elements of it were copied by some. The general concept of a GPMG was adopted by pretty much everyone postwar. The concept likely would have seen adoption regardless of the presence of the MG42 and the earlier MG34 that pioneered the concept, as we see with the British development of the Bren Gun that had similar aims (albeit worse execution). But there are issues that precluded simply adopting the MG42 itself.

The first and most obvious is ammunition compatibility. The MG42 was chambered in 8mm Mauser, and postwar operators would be using any number of different calibers. Conversion of the design is an option, but it's not trivial. The US, for example, had arguably the best chance with the similarly-sized 30-06 cartridge, but their experiments with rechambering MG42s were fraught with reliability issues and they never really got a simple conversion workable. The Soviets and British, both using rimmed rifle cartridges, would have an even harder time adapting the design to their indigenous cartridges. On top of that, there's the normal factors that slow procurement - the marginal improvement not being worth the cost of re-equipping. Armies at the end of WW2 were demobilizing and were already trained for the large stocks of equipment they had fought the war with. Wartime experience had already shown that the impact of German GPMGs was not decisive in infantry combat, and thus the limited efforts would focus on other areas.

For the Soviets, they seemed to focus on making firepower more compact and controllable with the 7.62x39 cartridge, which they would develop a rifle (SKS), submachinegun (AK-47) and squad automatic rifle (RPD) to fire. Though they didn't have a common platform for GPMG roles, their wartime weapons were generally good enough (SG-43 in the medium machinegun roles, DPM in the light machinegun roles), and the Soviets were content enough for some time with the belt-fed conversion of the DPM (RP-46) to mitigate the most obvious failings of their light machineguns. They did finally come around to adopting a true GPMG, but that came at the end of the 1950s after experience showed that their squad automatics (RPD and RPK) didn't provide the firepower they wanted. However, they would develop an indigenous design - the PK - that would in many ways come to be considered a gold standard for GPMGs going forward.

The US, meanwhile, had more pressing needs to replace their light machinegun - the BAR - which was probably the weakest of the major powers' guns, but they opted to develop designs indigenously. The result here - the M60 - drew heavily on successful elements of the MG42, but it had a more reasonable rate of fire. The rest of NATO, meanwhile, generally opted for the FN MAG, despite the development of an MG42 derivative that chambered the NATO standard cartridge (the MG3). There were plenty of reasons behind that, but a big deciding factor was the fire rate - the fire rate of the MG42 and its derivatives hits diminishing returns, especially in a time when the antiaircraft role of GPMGs has fallen out of favor. Now, the high fire rate is more of a liability due to excessive expenditure of ammunition and more rapid heating of the barrels.

Effectively, the GPMG concept exemplified by the MG42 was generally adopted worldwide, but in new designs rather than by adopting the MG42 itself. While the MG42 had excellent elements to it that were copied by some later designs (the top cover in the M60 and the feed mechanism and trigger in the FN MAG), conversions were less popular in the Cold War.