If you consider that Tolkien first wrote the book without clearly knowing himself exactly what was going to happen next, then the question is not as straightforward as it sounds.
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Saruman would probably utterly destroy Rohan (leaderless, as Theoden would remain under Grima's influence) and Sauron's assault on Minas Tirith would be succesful.
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Correct, given the circumstances in the finished book.
But if Tolkien had decided against Moria, and decided to take the long way round, what would have happened then?
We find out later that Théodred and Company are defending the Fords of Isen against Saruman. It is possible that had the company continued on through the Gap of Rohan they would have encountered Théodred instead of Éomer, and most of the rest of the Rohan story could be left intact but with small changes.
An encounter between Gandalf and Saruman would have been on the cards, I think. Perhaps Gandalf would even have ridden to Fangorn to rouse the Ents himself (he could have summoned Shadowfax by thought, as he did in the actual story). With all the opportunities for battle, Boromir could still have died defending Merry and Pippin. The question is, though, how and why would the Fellowship be broken? Tolkien himself stated that without the breaking of the company, the quest could not have been achieved.
It is likely that even if Gandalf and a few others stayed in Rohan to help with the fight against Saruman, Aragorn at least would have gone east with Frodo and Sam, desiring to go to Minas Tirith. The entire story would of course have been completely different, and the appearance of Gollum more questionable. It makes for some interesting speculation though!