Will USA Apple Watch S10 cellular work in Italy?

You should contact your Italian carrier, they should know if the different LTE bands are supported:

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To cover the LTE and UMTS bands used around the world, there are two separate cellular models for Apple Watch. They have been optimized for use in the country of purchase, and not all models work in all countries or regions. There is no single model that supports cellular service worldwide. To see which carriers in your country or region offer cellular service for Apple Watch, refer to the chart below.

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copied from: Apple Watch - Carriers - Apple

As you can see on the official cellular network page Apple Watch - Carriers - Apple

The US Ultra 2 model is the same in the US and Europe: Model A2622

I bought a US Ultra 2 back in October and use it with my eSim without trouble in Switzerland as well as I have used it in France.

Just make sure from that cellular page that your operator is mentioned in the list of Model A2622

It looks like there are 2 operators: TIM and Vodafone

Also, for now, the Oxygen device is disabled. But once the case is closed, Apple may enable it with a software update.

You may have warranty service or limited warranty service. Service will probably be limited to Apple Retail Store, as third party providers may refuse service. Here’s a map of Apple Retail Stores in Italy.

Warranty will be valid in the U.S., not Italy.

The U.S. O2 sensor restriction will not go away went the watch leaves the U.S.