Apologies to the OP, but I have had an alternate time-line in my head for a while, so thought I would share it.
What If: Robb Stark refuses to marry a Frey girl?
My thought is that Robb's ultimate objective is to rescue his father. So, he baulks at the thought of marrying a Frey girl and decides to push on towards King's Landing - hoping the Lannister army is tied up besieging Riverrun. Instead he encounters Tywin's host at the Green Fork. Unlike Roose Bolton, Robb doesn't march overnight and has a much bigger force than the original timeline. As a result Tywin's original plan works, the left flank (with Gregor Clegane and Tyrion) of the Lannister formation collapses, the Stark's try to flank on the left, however, Kevan Lannister, commanding the middle wheels his force around to push them into the river with support from Tywin bringing up the reserve. The consequence is that the Northern forces are pushed into the river with many many killed, drowned or captured.
If you want to add some extra flair to this timeline, perhaps the Frey's don't sit out the conflict but realise they need to pick a side. In my imagination they send a force out after Robb, either taking the Northern army by surprise in the rear, or by dispatching survivors fleeing the Lannister routing.
A few possible consequences:
1. Tyrion, Bronn or the Mountain are potentially killed when the left flank collapses.
If Tyrion dies, then Tywin's chances of survival later go up somewhat. However, with Tyrion not sent to King's Landing, potentially the naval battle goes much better for Stannis. On the other hand if Tywin goes straight to King's Landing after the battle, perhaps he is able to fortify the city in a similar or even better way.
Additionally, if Tyrion dies, he is no longer able to marry Sansa.
If Bronn dies, he is Tyrion's Hand (in effect) and puts Tyrion's plans to good effect, but if he dies potentially Tyrion could just replace him. On the other hand, if he can't replace him, then Bronn doesn't protect Sansa from being beaten, or assassinate Symon Silver Tongue who is blackmailing Tyrion or put up the chain that stops Stannis's fleet from escaping the wildfire. So, Tyrion has all these great ideas but can't implement them.
If the Mountain dies, then consequently he can't kill Oberyn Martell. He won't go on to become Ser Strong and Cersei may face a much more uncertain future as he can no longer defend her in trial by combat.
2. If the Stannis loses the Battle of the Blackwater then Sansa is replaced by Margery as future Queen.
In this scenario if Tyrion is still alive she may be married off to him to prevent her marriage to Willas Tyrell. However, I think it is more likely she is married to Willlas or Loras (as a reward to the Tyrells) and sent back North to reclaim Winterfell. Alternatively, she is married to an ambitious surviving Northern Lord (e.g. Roose Bolton) or a Manderly in return for the North's allegiance to the Iron Throne.
3. Tywin potentially still dies by crossbow - but is killed by Varys instead.
There is still plenty of chaos in the Seven Kingdoms. Stannis may still be alive and the Greyjoys are still in rebellion.
The caveat here is that with an easy victory over the North, and if everything else goes the Lannister way (Riverrun surrenders, Littlefinger brings in the Vale and the Reach, and if the Mountain dies and relations with the Dornish thaw). Also, with competent replacement hands like Kevan waiting in the wings, Varys may decide killing Tywin is ultimately pointless.
4. The Tully's
In this scenario I imagine Catelyn is either executed or perpetually held hostage somewhere. If the Frey's side with the crown I imagine they are probably made Lords Paramount of the Riverlands and Edmure is married off to a Frey girl. If the Frey's remain neutral I could see someone like Cersei being married to Edmure instead.
5. Purple Wedding, Myrcella betrothed to Tystane Martell probably still happen in some form or the other.
The Tyrells are still motivated to kill off Joffrey. Meanwhile, the Lannisters will likely still want secure the Dornish alliance via marriage.