Will Apple watch series 8 45mm bought in USA works with Indian carrier ?

Probably not. To cover the LTE and UMTS bands used around the world, there are two separate cellular models for Apple Watch Series 8, Apple Watch SE and Apple Watch Ultra. They have been optimised 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 network providers in your country or region offer cellular service for Apple Watch, refer to the chart below.

Apple Watch - Carriers - Apple

Main thing is No Apple Support and No Warranty for Watches bought outside your home country

If you use the same carrier then it might. You still are risking a lot of money with no warranty, or Apple Support if it goes wrong in warranty. Indian Apple repair shops are famous for wanting nothing to do with grey imports.

Lets hope you don't need help from your Apple Store. Good luck.