I'm not very interested in "last days" speculation, although I have a suspicion the anti-Christ is alive and well (for my money he's an American political figure, and not Trump although I don't have much admiration for his gigantic ego).
As for NATO leadership I suppose Germany would have a large say as they are right in the centre of Europe and have a strong military tradition.
In regard to the "ten kings" of Revelation fame, I'd look further afield The prediction is that the anti-Christ will have a world-wide dominion, not a revamped version of a European bound Roman Empire.
I think they'd be ten administrative regions around the globe - for example - North America, South Africa, Russia, Central Asia, China, India, East and South East Asia, Australian and Oceania, Europe, Africa - just for argument's sake.
With satellites, starlink, silicon chip implants, biometric data, the internet and the compulsory use of electronic funds such a scenario is quite posssible these days. All it would take to set the scene would be a catastrophe big enough for most people to accept dictatorship in return for secular security.
As for the Vatican taking over the world with its Swiss Guard, they'd need to wield their halberds with ferocious efficiency - all 135 of them!
When Churchill mentioned the Vatican, Stalin asked: “How many divisions has the Pope got?”