The city of Rome had many alliances within Italy which proved vital to their success.
Why did cities continue to be loyal to Rome when the war looked utterly hopeless (After Cannae)? Ratio wise, how many Italian cities switched sides?
Hannibal's cavalry commander Maharbal famously quipped "You know how to win, Hannibal, but you don't know how to use a victory." (Livy 22:51). Rome had suffered three major defeats at the hands of Hannibal, and it may seem like Rome's situation looked 'utterly hopeless' (as you say). That wasn't the case though. Well over 100.000 Roman men had died in the three military disasters, but Rome still had manpower reserves of more than 400.000 men. Rome was not defenseless.
Livy (22.57) says that a praetor was sent to Canusium (near Cannae) to take command of the survivors, while 1500 sailors were called to Rome to aid the defence and a naval legion was sent out as blocking force. Additionally, 4 emergency legions were raised in Rome, and 8000 slaves were armed. All this was organised immediately after the news of the defeat at Cannae became known. It would take at least three weeks for Hannibal to march on Rome from Cannae, so there was time to prepare a defense.
All in all there were over 30.000 armed men ready to defend Rome's walls. Walls which were a lot more formidable than those of Saguntum - the siege which had started the Second Punic War. For comparison, the siege of Saguntum had taken Hannibal 8 months to complete; it took place in his own backyard so to speak, with secure supply lines and no serious hostile relief forces in the area. By contrast, for the siege of Rome Hannibal would be deep in enemy territory, with no (let alone secure) supply lines, and with the prospect of relief forces showing up at any time - either from Roman allies or from recalled legions from Spain or Sicily.
So Rome's situation was not as utterly hopeless as it seemed. In any case, it was not Hannibal's goal to conquer Rome. Only once did Hannibal march on Rome (in 211), but that was for a strategic purpose: drawing Roman armies away from his ally Capua. Hannibal merely wanted to reduce Rome to just another city state in Italy. The way to achieve this was by taking away Rome's allies. By marching into the Roman heartland and by defeating Roman armies, Hannibal wanted to drive a wedge between Rome and its allies; if the allies decided to move away from Rome, they'd naturally turn to him. In turn, this would force Rome to accept peace terms which favoured Carthage.
How effective was Hannibal's strategy? Or, as you ask, how many cities switched sides? After Cannae many cities in southern Italy defected to Hannibal, the most important one being Capua. Some others were conquered by Hannibal. Other still, including the most important ports, remained in Roman hands. John Lazenby has calculated (based on Polybius and Livy) that by 212 BC 40% of allies were no longer available to Rome. Furthermore, in 209 BC 12 out of 30 Latin colonies (generally the closest allies of Rome) refused to supply men because of the large demands put upon them by Rome. Additionally, Hannibal by now had the largest city in Sicily on his side and had made an alliance with Macedon.
Still a lot of allies remained loyal to Rome. Especially the cities in central Italy, which were closer to Rome, felt it was in their interest to stay on Rome's side. As explained above, Rome wasn't utterly defeated yet. By the Second Punic War, it had been one or two centuries since these cities in central Italy had become under Rome's influence; the alliance was a lot more secure than it was during the First Punic War (when some allied soldiers had planned a rebellion against Rome - see Harris p.98). Rome and its allies had fought together against Gallic invasions - Gauls who were now fighting on Hannibal's side. Moreover, Hannibal's presence was likely to be temporary; Rome would be around far longer, even if in a reduced state. For the northern and central allies, it was in their own interest to stand with Rome.
To convince the remaining Roman allies to switch sides, Hannibal needed still more decisive victories. The Romans denied him this though. Following the Fabian example, they refused to fight him head on. Instead, they went after his allies and any detached or foraging forces. Meanwhile, the Romans took the fight to Spain and ravaged Hannibal's home base, eventually conquering it.
Hannibal's strategy proved to be ineffective. Before Cannae, he was free to roam Italy and damage Roman interests where-ever he liked; after Cannae, the cities which joined him expected his protection against Rome. Hannibal had to split his force to leave garrisons, and he was forced to stay in southern Italy. Instead of taking the fight to Rome, he now had his own territory to defend. In the end, he achieved very little in Italy. When the Spanish part of the Carthaginian empire was lost and Rome threatened the city of Carthage directly, Hannibal was finally recalled from Italy.
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