We're getting there. Remember that Russia has a GDP less than that of California. I'd say that their economy is heavily involved in their military effort.
Recall that the allies won the Cold War by basically bankrupting Russia. They aren't much more wealthy now, if at all. A great deal of whatever wealth they accumulate goes to kleptocrats and Putin (probably redundant). So yeah, they're in pretty deep. And that's just Ukraine. What do we know about their involvement in Iran, and if they plan to do more? And China? They're probably not heavily committed.....yet.
And now Trump has asked several of our allies to help keep the SoH open. Happily they are as of now refusing. Basically their take is the "you break it, you own it" way of thinking.
As for the "majority of the world's economies", how many were on board in WWII? I'd wager that many countries in Africa and SE Asia didn't have too much money invested. And today, a very few nations represent the vast majority of the world's wealth. So if the US, China, Russia, Europe and India are involved, that would be pretty much world wide involvement from an economic standpoint. Not saying that all those players are on board now; but they all have a dog in the hunt.
I would not call today a "world war". But then when Germany took over Poland it wasn't WWII; but it got there pretty soon after.
And the wheels are greased, greed, incompetence, miscalculations, etc. And no clear plan for putting the brakes on. And Trump driving the clown car.