What happens would be completely dependent on how big the eruption is. It's a monster that puts feet of ash across the Midwest, the USA now has a huge problem - and the force of a blast that size would demolish everything from the West Coast to the Mississippi River. Tens of millions of dead in the US alone, and their ability to fight the war would be completely shattered.
The Japanese, reliant on oil from America and have major weather changes as a result of the big one, are not gonna fight on either. Invading anything beyond the Phillippines is completely ASB - the resources are not there.
Germany is completely up a creek without a paddle. The Russian winter of 1941-42 caused them a lot of pain in OTL, and this one is gonna be a LOT colder and hasher. Any of their aircraft will be really in trouble with their engines - and won't be flying as a result.
Famine would grip most of the Northern Hemisphere. North America and Europe are already blasted to kingdom come, Russia will be freezing solid soon. The war slackens off badly.
Tens if not hundreds of millions of people go south as fast as they can. The US will probably outright annex Mexico, Cuba and some parts of Central America for themselves.
Britain's government moves to a British colony in Africa or India, so watch for parliament recovening in Bombay, Salisbury, New Delhi or perhaps Sydney, Melbourne or Cape Town. British people go south in numbers. Australia's population jumps massively, while the white populations of South Africa, Rhodesia and East Africa skyrocket. A big stack of British land in India, too. Canadians would have to go somewhere, too - they probably join the British exodus, and gather in South Africa and Australia. This exodus landing in South Africa dooms apartheid and the Afrikaners.
The French try to take back their colonies in North Africa but don't have much success at it. Most of the French leaving France land in West Africa or Madagascar.
A bunch of others go to South America, as well. Argentina and Brazil are long independent by WWII, but they gain substantially. Argentina (and Brazil to a lesser extent) would become one of the world's chief food growers after this.
The world changes completely. Germany may be a dominant European power, but that is only temporary - the Brits and French will return, probably by the late 1940s or early 1950s. The Southern Hemisphere, particularly Africa, will be much, MUCH better developed. Racism will exist for a long while, but the countries would likely gain independence if not along the lines of OTL not all that much later.