I take it your question is less about Roman logistics and more about the specific fact that the forces at Cannae were effective crushed as a standing army, and how did Rome recover from something like that?

First thing is first, with Cannae specifically there were very real and very profound consequences. Several city-states defected from Rome and turned to Carthage. Most estimates put Rome's total loss at over 70,000 with almost 40,000 of those being straight up casualties.

The Second Punic War devastated Rome.

So what was their response?

Well, for lack of a better term, conscription.

The Carthaginians attempts to parley after Cannae and it was rejected. Instead, Rome conscripted everyone they could get their hands on, including peasants with no ties to land, as well as slaves. Furthermore, Rome flexed its legendary resilience by quickly adapting military doctrine and tactics, developing ways to counter Hannibal's classic flanking technique and changing their strategy to never put that many people under one command again, instead relying on much smaller, independent forces to face future foes.

In short, Rome survived because it was never too proud or too stubborn to find a way to make things work. Making the best of what they could gather, and getting mad rather than despairing. They took on totally new tactics and exploited the weakness of Carthage; attacking everywhere Hannibal was not.

The Roman army at that time was half composed of Latin allies so those losses weren't just being borne by Rome alone. This meant Rome could draw on a much larger base of manpower than other states who could only depend on their own population.

That said, even Cannae was extremely distressing and Rome was being defended by old men and young boys. They even formed a legion out of freed slaves equipped with captured arms taken from the temples.

Lastly they gained a breathing space when Fabius Cunctator was appointed dictator and he did the incredibly un-Roman thing of not fighting. He'd just shadow Hannibal's army and remain on the higher ground (where it'd be disadvantageous for Hannibal to attack). This kept his army intact and gave it time to train and build up cohesion, it hampered Hannibal since he couldn't disperse his army to forage for supplies. This went on for a while allowing Rome to rebuild her armies and let the other armies in Spain and Sicily go on offensive campaigns.

Related question, it seemed that Rome the city itself was vulnerable to sacking after Cannae. When his subordinates suggested marching on Rome, why did Hannibal sat on that decision? Was he waiting on Carthage diplomatic manoeuvres?

They also hired their own elephants and deployed them effectively in battle against Macedon at the battle of Pydna, and later during the invasion of Britain.