We don't know Tobirama's age, but I think Tobirama would remain Hokage until the end of the 1st Great Ninja War, and then he would appoint Hiruzen as Third Hokage when peace is concluded, just like when Minato became the 4th Hokage after the end of the 3rd Great Shinobi War, by this move Tobirama as Hiruzen did in canon would promote a new era for Konoha, someone from the next generation, so nothing would change here with the original story.
I don't think Tobirama would have remained Hokage until the 2nd Great Ninja War as an old man, it doesn't fit with Tobirama's character whose ideology is to value the next generation, Prime Hiruzen as Hokage was the best option, letting Hiruzen rotting as a Jonin until his 40s wouldn't be fair.
And I am pretty sure that Tobirama being Hokage was an accident, Hashirama probably died at the beginning of the 1st Great Ninja War, in time of emergency Konoha needed a strong man, so the number 2 in command (Tobirama) became Hokage as Tobirama was considered as the strongest ninja of his era after Hashirama's death. (stated by an Anbu guy + in the anime when Mu said that they need to talk to Tobirama because he believed it was him who summoned them, the 2nd Mizukage had a fearing reaction which says a lot about Tobirama's reputation in that era [EPISODE 267] )
Hashirama and Tobirama probably planned to make Hiruzen the 2nd Hokage, but thing didn't go as expected as Hashirama prematurely died in time of war.
The thing that would change in the plot, would be that Tobirama would be Hiruzen's advisor, so Konoha would be better led, with Tobirama around Danzo would remain loyal, and Tobirama might have trained the young Orochimaru. I think Tobirama would notice Orochimaru's great talent and would have tried to train / guide him, he would have noticed Orochimaru's darkness as well, I don't know if Tobirama would have been able to prevent Orochimaru becoming a villain but what I am sure is that contrary to Hiruzen he wouldn't pretend nothing is wrong with Orochimaru and might try to guide him. We know that even as a villain, Orochimaru considered Tobirama as the man he respects the most, so maybe Tobirama would have been a grandpa figure teaching to Orochimaru everything he knew about ninjutsu and ninja stuff, which Orochimaru would enjoy a lot. Hiruzen as Hokage didn't have time to enjoy time with Orochimaru who was orphan (just like Naruto and many other people at Konoha) while as a retired Hokage Tobirama would have time to do so.
Tobirama would stay alive until the 2nd Great Shinobi War when the Sannin become adults.