apple watch outside of US

If you bought the GPS only model, it does not use your cellular carrier at all. It will work like a GPS watch works regardless of where you are in the world. It cannot connect to a cellphone provider anywhere, not even the U.S.

If you bought the GPS+Cellular model, then it will not roam when outside of the U.S, and will not be able to be connected to local carriers outside the North American region. See link provided my Limnos in his reply.

Apple Watch - Carriers - Apple - https://www.apple.com/watch/cellular/

"To cover the LTE and UMTS bands used around the world, there are two separate models for Apple Watch Series 5, and three for Apple Watch Series 3.2. 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 roaming worldwide. To see which carriers in your country or region offer cellular service for Apple Watch, refer to the chart below." Click on the link to read more.

The GPS only Apple Watch can do Everything the Cellular model can do AS LONG AS your iPhone is nearby, where nearby is generally somewhere in the same moderately sized home. The Cellular model allows you to leave your iPhone at home while you go for a walk, run, bike ride.

So if the iPhone works, the Apple Watch GPS only will work.

As Limnos says, if you have a Cellular Apple Watch, then A) the Apple Watch does not do cellular roaming, so as soon as you leave your iPhone's carrier's coverage, the cellular Apple Watch reverts to behaving like a GPS only Apple Watch.

B) if you get your iPhone a new cellular service in the new country, then if that country is Canada, Mexico or Puerto Rico, then if that carrier supports the Apple Watch, you can use the Apple Watch as a cellular Apple Watch. If it is some other country, then the U.S. cellular radio will not work, and the Apple Watch will again behave as if it was a GPS only Apple Watch