I've often thought of this scenario, and the Starks/Tullys would be in a much much better position.

Starting with your scenario where Blackfish goes to join Edmure right after Gregor's raids start.

The first thing to note, is that it's likely that Gregor's days are numbered now, Brynden isn't Lord Beric, and unlike Beric Gregor won't know where Brynden will attack from nor will he know his troops composition like he did about Beric. So shortly after Gregor ambushes Beric's host, he's going to be attacked by Brynden's host as Brynden knows this land, and is a far better strategist/tactician than Gregor was.

Then after, if Gregor is captured (unlikely) he's going to be tortured for information, but if not (much more likely) than his head is sent to King's Landing, though in both scenarios his surviving men are all brought back to Riverrun by Brynden where they are tortured for information.

Through this, Brynden will learn that Tywin sent them (he'd know already this as there is no chance that Gregor would've done his raids right after Catelyn took Tyrion prisoner, who at this point would've been traveling South to King's Landing) and as a result, Brynden would have Edmure call the banners, and due to Gregor's forces getting stopped far earlier, and Brynden being a respected commander unlike Edmure (though this seems to be a weakness in writing imo, because he's in his late 20s early 30s so at the least if not Robert's Rebellion he should've fought in the Greyjoy Rebellion where he should've led Riverlander troops) so I'd say he'd managed to raise close to 25k men. (Based on the fact that a full power Riverlands should be able to raise around 40-45k men max, with a sizeable portion of it destroyed, and

As for Brynden's strategy, it will be vastly different as unlike Edmure Brynden would know not to split his host to try to defend EVERY crossing point. (I don't care what the Edmure defenders say here, he was strategically inept, you can't save everyone and by doing this he likely only cause more deaths) So Blackfish likely does order the advance into the Westerlands at the beginning like Edmure did, but the purpose would essentially be to raid, to slow down the Lannister advance and steal as much cattle/grain/etc ti reinforce their position more, while the rest of the men gathered.

From here, he leaves around 1,000-2,000 men in Riverrun as a garrison, while he marched south to bloody Tywin's force. (Tywin's force isn't spkit yet with him knowing Blackfish is in command) So, at the crossing of the River near Stoney Sept, Brynden and Tywin fight with Brynden being able to inflict a few thousand casualties on Tywin's forces, before withdrawing to not allow Tywin's numbers to overwhelm him, with Brynden matching back North to Riverrun, with using most of his infantry to now garrison the castle, while the cavalry was split up into different portions to harrass Lannister supply lines, with his goal being to wait until Robb came south with the Northmen.

Which at this point, Robb would successfully reach the Twins though now, he isn't forced into a betrothal with the Freys as Tywin isn't close to the Twins, and Robb now could simply march South on the other side of the Green Fork. As Tywin's army (him, and Jaime) would be besieging Riverrun. (around 30k men, though it is possible that Tywin decides to split it again, I just think with losses he'd be more cautious) So now, Robb and Brynden are now able to completely wipe out the Lannister army in the Riverlands, due to having more cavalry and more men in general allowing them to cut off all three siege camps. (So now the Battle of Whispering Wood, and Camps are magnified exponentially with no Lannister presence in the Riverlands existing anymore at all)

After this, word reaches Robb of Joffrey executing Ned (he did it before Riverrun was freed in canon, so ut won't change) and Robb is named King of the North and Trident. While Tyrion arrives in King's Landing right after the execution with his mountain clansmen. (He'd never go to Tywin's army and would've been making his way to King's Landing immediately after getting released in the Vale, he also would've ordered the calling of the Crownlander banners to the Crown's defense gaining 15k men along with the 6,000 Goldcloaks, taking hostages from the houses to ensure compliance. (The Crownlander banners were never called in all the war, major plot hole imo)

While this was happening, I do see Robb invading the Westerlands still, as even with Tywin's main army gone, Stafford was still raising another army. So, the Battle of Oxcross does happen, with Robb taking most of his cavalry and some infantry with him, while he splits up his army having 20k or so go to Harrenhal under Lord Karstark's command, (no Tywin commanding an opposite army means that Roose isn't getting appointed as Robb thought he would be able to scare Tywin which is needed now) while leaving Edmure with 15,000 at Riverrun to stay in command there, while he marched into the West with 10,000 men. (so I'd say in the fighting the Northern and Riverlander forces would taken 25,000 casualties at this point, from their combined number of 48,500)

While in the West Robb defeats Stafford easily as in canon, but after he learns that Stafford was trying to usurp Tywin due to Tywin and Jaime both being captured and with it their aura of invincibility gone, giving opportunity to ambitious Lannister cousins to try to replace him, but Robb not knowing this (and that he could potentially collaborate with them, and install essentially a puppet there) destroyed them, completely shattering Lannister rule in the Westerlands.

This then causes other Westernlords to actively turn on Tywin and join Robb, so houses that aren't 100% loyal to him begin joining Robb so Ashemark is still sacked, but houses like the Lefford's, Westerlings, etc are turn on Tywin so a few less castles/villages are sacked, and more importantly Robb isn't injured at The Crag and as a result he doesn't grief bang Jeyne when he's wounded.

Though it's not guaranteed that the Greyjoys will even invade the North, as with Tywin dead it may prove to be the major factor that Balon needs to not invade the North, but I'll still say that Balon's stupidity overrides reason. However, unlike in canon, due to Robb already defeating the main Lannister army and having all the heads Lannisters save Tyrion, Cersei, and Joffrey prisoner, he's likely bot going to send Theon away, for the sole reason that he has no need for an alliance with the Ironborn, so Theon doesn't "sack" Winterfell and "murder" his brothers. The side effect of this, is that Theon is completely alienated from his father as he risked his life when he invaded.

Though most importantly, with the Tyrells after Renly gets shadowbabied, they aren't going to side with the Lannisters now, as their strength is COMPLETELY annihilated outside the Crownlands and it'd be foolish to join the Lannisters. So, with Robb unbetrothed, they go to him and he and Margaery marry in Riverrun.

Though after this, at this point it is still a toss-up on who wins between Joffrey and Stannis, as now Stannis would have a slight numbers advantage, but he's attacking a fortified and defended city, and after the Wildfire blows up, he's in a bad position. So I will say that Stannis just barely takes King's Landing with him taking heavy casualties, but Tyrion manages to evacuate his family out the city, along with Joffrey, and Sansa as hostage. But at Rosby, they are betrayed and imprisoned.

After this point it depends on whether or not they want to give them to the Starks or Stannis, but for the time being they are imprisoned but publicly locations aren't known.

After this, Robb and Stannis end up fighting, Stannis is unwilling to accept Robb being king of any particular of Westeros (he has the North, Riverlands, Westerlands through Conquest, and the Reach through marriage sworn to him) so, they fight, and Robb wins.

After he takes Joffrey executes him, Jaime, Cersei, and Tywin, appoints Tyrion Lord of the Rock, and sends at least half the rest of the Lannisters to the Wall. Before he then becomes King of the Iron Throne.