In general, nations retained the same cartridges used in ww1 up until and through ww2. Like with bolt-action rifles rather than semi-automatic ones, this was mostly due to cost during the interwar years, and then the need to maintain production of arms and ammunition during the war.

Several countries realised the limitations of the ww1 rifle calibers and tested or even attempted to introduce new ammunition and/or new service rifles. The British tested .256 (6,65mm) and .276 (7mm) ammunition and the French started a switch from 8x50Rmm to 7,5x54mm. The Italians actually attempted to switch from 6,5x52mm to 7,35x51mm in 1938, but the project did not get far before the war broke out. The US also tested the .276 (7x51mm) Pedersen cartridge and actually adopted it for the new M1 Garand service rifle, but reversed the decision after verifying that the Garand would work with hte heavier .30-06 cartridge.

There were extensive stocks of earlier rifle caliber weapons left from ww1 - rifles, LMGs and HMGs alike. Adding to that were the fact that production was tooled and streamlined for the older cartridge - setting up new production lines are usually both expensive and time-consuming meant that the older cartridges had to soldier on.

Lack of money to re-tool production lines before the war, with demobilisation and then the Great Depression throwing a monkey wrench into military re-armament plans and then the urgent need for arms and ammunition in the field once the war had started prevented most of these projects.

Instead all sides attempted to improve ballistics and reduce recoil by introducing lighter bullets, spitzer bullets and otherwise improve existing cartridges.

Vague concepts like stopping power played a part, as did concerns over the maximum effective range of machineguns and commonality of rifle and machinegun ammunition. In general, though, it's very much a case-by-case basis, and we see several major powers make moves to adopt smaller-caliber cartridges before or immediately after WW1.

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Japan's transition away from the 6.5mm cartridge - which to modern eyes seems ideal - came as a result of experience in China. Unfortunately for Japan, the war in China taught them a whole list of wrong lessons, and among them was the idea that the .303-British-derived 7.7mm machinegun cartridge then in service with some of their heavy machineguns was superior to their 6.5 mm cartridge. The 7.7mm round, as with the German-supplied 8mm machineguns the Japanese encountered, was found to have better performance at long ranges compared to the 6.5mm round, and the added power was considered desirable. Given that light machineguns then in service - the Type 11 and Type 96 - were chambered in the same caliber as the Type 38 infantry rifle, it provided more impetus to switch the entire supply chain to the 7.7mm round. Doing so would simplify logistics and, with the introduction of the Type 99 Rifle and Light Machinegun, continue the practice of allowing riflemen and machinegunners to share ammunition.

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Italy's situation was a bit different - they were seriously lagging behind in ammunition design, still using a heavy bottle-nosed bullet though WW1. The decision early on to use gain-twist rifling that was tailored specifically to the bottlenose bullet 6.5 Carcano was originally designed with helped improve barrel lives and accuracy with the round, but had the side effect of being poorly-suited to use with more modern spitzer (pointed) bullets. Because a transition to spitzer bullets would require re-tooling every factory to use more conventional rifling methods, Italy continued to use bottlenose bullets while every other major nation had already transitioned to spitzer rounds. That left Italy with most of the downsides of small-bore ammunition - shorter range and less power - while the poor aerodynamics and heavy weight of the bullet would further hurt performance at a distance and eliminate much of the flat-trajectory advantage most 6.5 rounds have. Following poor performance in Ethiopia and Libya during the interwar period, Italy finally made an attempt to modernize its inventory. Along with a very modern and utilitarian new rifle, Italy necked up the 6.5mm case to fit a 7.35mm spitzer bullet that was lighter than the 6.5mm round. Here, the result was a more modern cartridge with better aerodynamics, a longer range, and (supposedly) better wounding potential. Unfortunately, this transition started in 1938 and was very quickly cut short by the outbreak of WW2.

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Outside of those two prominent cases, however, we see the US and Britain both make moves to adopt ~.28 caliber or 7mm rounds at different times (twice in the case of Britain), and France would decide to shift down in size from an 8 mm round to a 7.5 mm round. For the US, the .276 Pedersen cartridge was experimented with and considered for adoption when developing and testing what became the M1 Garand rifle, but it was decided that the logistical strain of switching to an entirely new cartridge was too much at a time when 30-06 was plentiful and Army budgets were limited. For Britain, they had similar designs as the Italians had between the wars in the aftermath of the Boer War - they wanted a modern new rifle with a modern cartridge to replace the flawed Lee-Enfield and its rimmed .303 cartridge, and they developed a Mauser-derived rifle and a .276 cartridge (7mm) to go with it. Unfortunately, the Pattern 13 rifle only came about in 1913 and had to be dropped due to the outbreak of war. It would be rechambered in both .303 British (Pattern 14) and 30-06 (Model 1917) during WW1 for Britain and the US, but it wouldn't make a comeback in the postwar drawdown. The second British 7 mm cartridge attempt happened after WW2 with the development of the .280 cartridge. While the cartridge itself performed well, it was met with resistance from US officials over concerns of power and range (particularly in machineguns) and ultimately was dropped in favor of standardizing NATO over the 7.62x51 NATO cartridge.

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France had the only successful transition, and that was probably because they had the most glaringly obsolete cartridge. Like the rifle it was designed for, the 8mm Lebel cartridge was a horribly rushed and flawed design, being a heavily tapered rimmed round necked down from the 11mm Gras cartridge. It wasn't particularly powerful compared to its contemporaries, and, as the infamous Chauchat machinegun magazine demonstrates, its heavily tapered design led to issues with the design of magazines. Post-WW1, France made steps towards creating a modern set of weapons, complete with a new cartridge. As this cartridge had to meet both the range and power requirements of the light and heavy machineguns and the performance requirements of the standard rifleman, they ended up with the 7.5 French cartridge - a design roughly comparable to the 7.62 NATO cartridge, but with a lighter bullet and a little less power.