How important was the war against Vercingetorix to the Romans?

The question is interesting, and I want to highlight some elements at the beginning:

• The Romans were called in Gaul to defend some people against others, and they used that first step across the Alps to take more territories: thus, the Romans felt like they were coming in with good right for them
• The war against Vercingetorix was caused by an uprising against the new possessions of Rome: thus, it appeared to the Romans they had the good right as well

Now about the specific question: At the battle of Alesia, the Romans had the defeat of Gergovia to revenge and they feel threatened by the possibility of an unified Gaul under Vercingetorix. Brennus was still a legend in each Roman mind.

So yes: the loss of that area of Gaul would have been seen as unacceptable to Rome and its authority and a new campaign to retake what was lost, would have been mounted as soon as possible.

It's a bit of both. Julius Caesar was under much pressure from his enemies in the Senate, who tried to use anything remotely possible to impeach him. A defeat against Vercingetorix would have been the perfect excuse to recall him, and have him trialled and banished.

The battle of Alesia was not that long after Julius Caesar took Germanic tribal leaders prisoner. According to Caesar, they were stalling for time. As it was, they came under a truce, which Caesar broke. The Senate (his enemies, that is) tried to make a big deal out of it.

That does not want to say that Vercingetorix was in the clear, or off the hook. Far from it! All that would happen was Caesar being replaced by someone else, to finish the job.

The Gallic Wars were justified wars, according to the Romans. The Adui had asked for help, and got it. Then the Germans had to be defeated, again, on the request of Gaulish tribes. Vercingetorix may have felt justified to wage war against the Romans, the reverse was true as well.

If Vercingetorix defeated Caesar's replacement, more troops plus another general would be sent to snuff out that embarrassment. The Romans didn't take defeat kindly.

On it's own probably not such a great deal. Caesar's invasion of Gaul had already grossly exceeded the goal of safeguarding the allied Aedui, the purported reason for the invasion in 58bc. The combined Gallic host under the Helvetii were roundly defeated at Bibracte and the much reduced survivors forced back to their homeland.

After eight years of campaigning in Gaul resulting in a pretty much one-sided routing of the Gallic tribes with resultant massacres of civilians and huge numbers of Gauls sold into slavery, the Gauls were significantly reduced as a potential military threat. During the entire campaign there is no sense that the Gauls of Gallia Narbonensis, a province the Romans had conquered over 60 years before Caesar decided to help the Aedui, caused any problems. The Gauls of northern Italy had long ceased to be an issue and Caesar granted the whole region Roman citizenship status in 49bc.

The whole campaign north was really a dog and pony show to fill Caesar's coffers and give him some much sought after power. If he'd been killed at Alesia then it's unlikely the Romans would have launched a major offensive to avenge him but they may have launched a punitive expedition against the Arverni resulting in hostages, tribute or similar and lands given to the Roman friend the Aedui.

Of course Alesia was in 52bc which means it came the year after Crassus marched an entire Roman army of seven legions to their death in Parthia at the Battle of Carrhae. Having already lost seven legions and around 8000 auxiliaries, potentially losing the ten legions under Caesar's command and a further 10,000 auxiliaries would be a major blow to the Roman Republic. Two such defeats back to back might even inspire uprisings in Hispania, Illyria and the Greek world. As the battle of Alesia involved the Roman forces being completely surrounded by their own defensive works, a sudden escape for them would be difficult if the battle had been lost, it's likely most of the Roman army would have been killed, perhaps even the Gallic allies. Any survivors would have to travel over 200km to find anything approaching friendly lines.