In your AGOT Cat V review, you said that, had Cat not taken Tyrion as hostage, she could’ve warned the North to mobilize for war before the Lannisters. Bryndenbfish argues in the 2nd part of his military analysis of Tywin that Tywin was already preparing for war at this point, considering the size of his army, and used Tyrion’s arrest as an excuse to start it. Don’t you believe that? Maybe Tywin’s numbers were a mistake from GRRM’s part? But Stannis was an imminent threat, so he could be right.

Respectfully, I think Bryndenbfish is wrong on this point. News arrives of:

“the Kingslayer is massing a host at Casterly Rock,“ Ser Rodrik Cassel answered from the room behind her. "Your brother writes that he has sent riders to the Rock, demanding that Lord Tywin proclaim his intent, but he has had no answer.”

in Catelyn VII of AGOT, which takes place 5 weeks after Cat V took place. If the Lannisters are still in mobilizing at this point (the Battle of the Mummer’s Ford, the first time that Tywin’s host takes the field in the War of Five Kings, takes place about 50 days after Cat V), then I don’t think it’s likely that Tywin has already begun mobilizing in advance of Tyrion’s arrest. 

To circle around to the main point, if Catelyn hadn’t been diverted to the Eyrie, her messages to Robb would have had a major head-start, potentially allowing the North to get its forces down earlier (remember, in OTL Robb’s army departs Moat Cailin about 3 days after Edmure loses at Riverrun, so a matter of weeks could make a huge difference).