Humanity would have steam rolled everything else.
Yawn.
During the great crusade the Eldar were still in post-fall shock, the tau were trapped behind a warp storm and the necrons and tyranids weren't even players yet.
Regardless of the Emperor's status, the Space Marine legions had been gutted and divided and Roboute Guilliman's reforms would have still have crippled the rest.
The manufacturing and military capabilities of the imperium wouldn't have magically kicked into perfection simply because golden boy was still kicking. Mars had been gutted, the imperium had been kicked in the teeth and it still would've been a limping recovery.
Of course they wouldn't have Magically won the battle against the Tau- it was primarily an air battle so even having the Big Shiny on the front lines wouldn't have changed things much.
Diplomacy might have gone better except that Emps was a notable xenophobe, so it wouldn't have happened anyway.
The two single biggest changes I can say could have tipped the balance is:
1. The second founding wouldn't have failed, and might have happened earlier
(If it had even happened. The high lords voted for it but Emps might not have)
So that's a bundle of extra space Marines which could have helped the imperium (or gone chaotic and caused more mayhem)
2. The emperor, whilst not being as good a beurocrat as the high lords of Terra, certainly was a more decisive leader, so certain events such as the Badab campaign may not have happened.
At the end of the day he was a pretty flawed person, so the amount of emotional trauma even from the heresy could have been a blow beyond telling anyway.
Maybe he's not dead, just sulking?