Will cellular work in India if we purchased in US?

iPhone Cellular Carriers that support the Apple Watch, which models, the radio frequencies each model uses, and the country where they offer service:

https://www.apple.com/watch/cellular

Also be aware the warranty is not international, so you would also have to return to the country where purchased for warranty repairs, or pay out-of-pocket.

The Series 4 and Series 5 ECG app also has country restrictions. The Apple Watch Must be purchased on a country where the government has approved the ECG app And it can only be used in a country where the government has approved the ECG app. Both must be true.

Apple Watch ECG countries

https://www.apple.com/watchos/feature-availability/#branded-ecg

The U.S. and India have approved the ECG app, but of course cellular will not work. So if you decide to buy the Apple Watch in another country on your way to India, make sure that country's government has approved the ECG app. Apparently Dubai is a very common stop between the U.S. and India, and many many users from India have complained that ECG doesn't work when they try to set it up in India, because the UAE/Dubai government has not approved the ECG app.

Finally, just so you know. The Apple Watch GPS only can do  Everything  the cellular model can do  as long as  your iPhone is nearby, where nearby is somewhere in the same 3-4 bedroom home (assuming typical wood frame construction).

Not on cellular? You will not be able to activate a US purchased Apple Watch cellular with a cellular network in India. The Watches are regional devices and that Watch will not activate in India. While it will work like a GPS only watch, there is no sense in purchasing a more expensive cellular watch and not be able to activate it. You will have to purchase a cellular version of the Watch in India.