Re: "Same day apple watch battery replacement?"
We are all Apple users, like yourself, in the community ... so no-one here can make promises for the particular Apple store you will visit. Community members' experiences will likely vary in the various countries and in different stores.
Suggesting you browse Apple's Apple Watch Service and Repair - Apple Support
web-page.
If you wish: You can scroll down the webpage to get an estimate, and / or click
the blue "Get service" button to check out your options.
Else: you could just phone the store to set up an appointment, and ask for any details you need. ( If needed: To Search for an Apple Store by location:
(You may need to enter your country at the top)
Store locator: Apple Store - Find a Store - Apple