I’m in Italy and my universal adapter charges my iPhone but not my AirPod case

If it works in UK it should work in Italy (the only difference is the physical shape of the wall socket) and if it charges your iPhone you have proved the power adaptor and cable are good. Your AirPods case will take less power than the iPhone, so that is also not a problem.

The likely suspect might be debris in the Lightning port for your AirPods case, preventing the connector from making proper contact. You can look into the socket, and if you see debris try to remove it as noted below. Warning - do not use hard or metallic objects like a needle in the charging port, this could cause permanent damage to the contacts in the case or even a short circuit.

If your AirPods won’t charge - Apple Support

How to clean your AirPods, AirPods Pro, AirPods Max, and EarPods - Apple Support