Larger Italian cities — Rome, Milan, and Turin among them — have dedicated 24-hour animal hospitals ('pronto soccorso veterinario') that operate independently of the on-call rotation system used in smaller towns, meaning there's always a vet on site rather than one reachable only by finding the current on-duty rotation. University veterinary hospitals, like the one at the University of Padua, also run round-the-clock triage and emergency services for companion animals.
The practical move is preparation before travel: search 'pronto soccorso veterinario' plus the name of each city on the itinerary, and save two or three results with phone numbers before leaving home. An app like VetFinder is worth having installed too, since it's built specifically to help locate vets — including English-speaking ones — anywhere in the world.