And some thoughts on the other big two Entente powers.
The Russian Empire is both easy and difficult. Simply put, what barrels? I don’t see the Russian Empire ITTL having the capacity to develop barrels to the degree that the OTL Soviet Union did. I see war-damaged, socially unstable Russia as being two steps away from OTL’s warlord China. I simply don’t see the Russians having any domestic tank industry (which frees up Soviet designs for other countries to use, like the Yankees).
However, I do see them having quite a bit of foreign-built armor and domestic armored cars. The latter would mostly be what the Russians used in WWI or what the Soviets had in the interwar period. The former would be Entente surplus sold/gifted to the Russians during and after the Russian Civil War: a mix of WWI designs (the FT-17 comes to mind immediately) and interwar designs. Perhaps some Confederate barrel designs make their way to Russia?
The situation in Russia is like OTL China: a hodgepodge of foreign designs slapped onto an ill-equipped army. I could see Russian infantry having helmets from all sides, mixed with WWI and civil war-era caps. Their main rifle is the trusty Mosin-Nagant, followed closely by pretending to have a rifle. They’re still hauling around water-cooled Vickers machine guns on wheels, or on a horse-drawn cart. However, like the OTL Chinese, they have a small corps of elite, well-equipped troops trained by a foreign power. Nazi Germany in the case of OTL China, and likely Actionist France in the case of Russia. This unit will likely be responsible for the only Russian successes early on in the war.
As for the British, I suspect that they get late into the war footing game. While the French are gearing up for round two as soon as possible, the immediate post-war mood in Britain is that of “isolation is splendid and the alternative is anything but.” They let themselves get dragged into a continental war and it got them nothing but some graves in Belgium. Until Mosley’s boys get into power, I think the focus of the Brits would be to maintain the empire they have left. This means they will be focused predominantly on keeping up with the Kaiser in naval affairs, ensuring the Japanese don’t have any creative ideas for East Asia, and reminding natives in the colonies that the old ditty about Maxim guns still applies. So, British interwar tanks would mostly be light, intended to operate in bad terrain against enemies who have no armor whatsoever. The design philosophy is something I see mirroring OTL Japan: tankettes and light tanks intended to support infantry.
When Mosley takes power, and especially after Spain, the focus shifts back to a war in Europe. This follows French and Confederate philosophy, two powers who were leaps and bounds ahead of the British in land warfare. Ironically, I foresee British designs being inspired by Confederate designs and in this way resembling OTL British designs.[1] However, I also think that “perfidious Albion” is more than happy to let the French do most of the dying, so they don’t really have it in mind to create massive expeditionary forces like they eventually did in the Great War. Instead, the British Army would be like it was in the beginning of the Great War: a small number of elite troops. Their barrels would be intended to operate in tandem with infantry in what I could only describe as modern combined arms tactics. Unlike the Actionists or the OTL British, TTL’s British don’t have an infantry/cruiser mindset. In fact, I see them pioneering the concept of the main battle tank (possibly called a “universal tank” ITTL, as it’s appropriately British-sounding; perhaps changed to “universal barrel” after the war, tank having the connotation that panzer does IOTL).
[1] We have to go deeper, Leo!