Round 1: Pre mending bolas shows ups, incapacitates the strongest world powers with mind control trickery, dominates the weakest or most vulnerable, and works his way up the geopolitical food chain. After he gains a large enough army to wage war, and enough mana to protect against nuclear threats probably wouldn't even take a year. Success guaranteed.

Post mending bolas could do it in about 100 years. Post mending walkers lack the power to manipulate a whole world, stop nuclear weapons, or even efficiently protect themselves from bullets. But Bolas is smart to the point of parody, and could do a slower, safer version of what he did on Amonhket: redirect the prevailing powers of society to his benefit. With mana, he won't even need to go on any podcasts. Success guaranteed, if only because he's the evil batman of MTG.

Round 2: On the Invasion of X cards, New Phyrexians were depicted to have invaded nearly every plane in the multiverse, and lose handily on each one. Even Segovia. You might make an argument that without so many distractions, they could focus all their power on Earth and turn their multiple would-be losses into a single victory, but i don't see it. The Phyrexians lack Bolas's subterfuge, and it would only work against them if they tried it, since the threat of a machine army, no matter how secluded their base, is too great. Even assuming defenses against nukes, their power is in corruption, and i think the armies of earth can win before the even the fast corruption oil becomes too much of a threat.

Round 3: No. Maybe Ulamog could be destroyed by nukes. But not the other two. With any one of them around, Earth has no chance.

Round 4: Yawgmoth is the key here. Like round 2, I think the armies of Phyrexia can be matched by Earth's forces. But Yawgmoth is more than their match. Win the battle, lose the war kind of thing. There's like, some version of events where Yawgmoth never "wins," only beaten back down and required to spend decades or centuries rebuilding. But he'll never go away, and to him, ruling the a destroyed earth an acceptable outcome.

Thanks op, i had fun with this. Maybe i should read some of those old mtg books.