So Pompey presumably beats Caesar after waiting a bit longer. The POD is that Pompey points out that he has "Magnus" in his name for a reason, and he is in command of the Legions, and the Senators, being unable to change things, accept his decision.
So now the question is what happens next? Caesar and Antony were both at the Battle of Pharsalus, so would either die in the battle, be captured, or retreat after the battle. The core of Caesar's army, his veterans from Gaul, was at Pharsalus, so he is going to have to raise new troops. Caesar had pacificied Spain and he still controlled Gaul, so he could retreat west, and try to raise new troops there.
Caesar could be finished though. He might be able to offer further resistance, but with the loss at Pharsalus he is going to have a tough time. The Optimates split OTL, with Pompey leaving the Republic and going to Egypt to try and raise new funds and troops, and others to Africa to continue the rebellion. Basically I think the situation is desperate for whomever losses the battle.
So in 48-47 Pompey returns to Rome and retakes the city. Many of the Senators and other prominent supporters of Caesar beg the forgiveness of the victorious Optimate faction. Caesar is in Gaul and Antony in Spain, trying to raise new Legions to continue the Civil War.
Now does Pompey (who would be elected Dictator once he retakes Rome) initiate a round of proscriptions in order to fund the final campaign against Caesar (and destroy Caesar's base of support in Rome if Pompey suffers any reverses)? If he does proscribe, then is the young Gaius Thurinus, Caesar's grandnephew, among those who die?
If Pompey wins the civil war with Caesar, then I think that he would follow in the footsteps of Sulla, and support the Optimates in their vision of the Roman Republic. This is going to mean the Republic gets to limp on for a few years more as the Republic, without OTL rounds of war between the pro and anti Caesar forces.
If Gaius Thurinus is allowed to live, then I think that he will end up as the leaders of the Populares faction. He is the logical heir of Caesar, and was a keen political animal. Even without Caesar's inheritance, he will have Caesar's legacy as a champion of the Populares faction.
In order for Thurinus to get to a position where he can be an effective leader however, he is going to have to get a provincial command where he can do some foreign conquest. Perhaps Thurinus become Brittanus or Germanus?