If Edmure manages a sucssesful defense against the Westermen in the first place, would he or the Riverlords accept Robb as their King? They were not at the brink of collapse in this scenario and are not as dependend to the North as in OTL.

Interesting question!

I mean, if Edmure had managed a successful defense against the Westermen, lots of things would change significantly.

For example, let’s say Edmure arranges a reverse-Red Fork and executes competently, as he did in the OTL battle, and even manages to refuse his flank vs. Tywin’s southern flank.

A couple things change immediately: 

1. There’s a lot more Riverlander troops around. There’s 4,000 from Vance and Piper’s doomed army, plus however many were lost at the disaster under the walls of Riverrun. How many that is something of a puzzle: I’ve made a few estimates, but I think another 5,000 may have been lost. Which means that the Riverlander army is going to be as big as Robb’s army at 19-20,000.
2. Lannister strategy is going to have to respond to their failure to break through. My guess is that part of their army is going to have to keep Edmure pinned down defending the Red Fork/Riverrun, while the rest is going to keep pushing east along the Gold Road, either trying to find an open way in to Edmure’s underbelly, or once Robert dies, shifting to defend King’s Landing from Renly or Stannis. 

This in turn changes Robb’s strategy immediately, to begin with, he doesn’t need to figure out a way to defeat two armies with one army, which required the splitting of his forces and the deal with the Freys. Yes, Edmure’s going to be demanding support rather loudly, as I would imagine Tywin would keep at least 15,000 battering away at the Red Fork, but it’s not as urgent a crisis as to require more than half of his army.

Instead, I would imagine that Robb would concentrate on moving south quickly – first, it gets his army closer to King’s Landing where his father and sisters are, second, it puts him closer to dealing with that southern Lannister army which threatens to outflank the Riverlander lines and put his allies on the run, and third that Lannister army might give him the Lannister hostages he needs to achieve his political aims. 

So what you could see happening is a major battle between Robb’s full(er) army and the southern Lannister army, probably being fought not that far from where the Fishfeed was fought. And if Robb can catch the Lannisters with their back to the Godseye or its river, he could achieve a tremendous victory that might, due to to its proximity to King’s Landing, actually deter his father’s execution.