How does Apple's Battery Replacement program work?

Don't call a store. There's no direct customer connection to an Apple Store. They don't allow it since they'd probably get swamped with calls. It goes straight to AppleCare once you ask for anything more than automated stuff like store hours.

You can stop in to make an appointment. In my experience that allows for a two-week window.

You can also make an appointment online up to a one-week window. Here's my way of doing it (for the US - don't know how similar it is for other countries):

• Start here: https://support.apple.com/iphone/repair
• Click on "Start your repair request" (blue button at the bottom).
• Click on "Battery, Power & Charging".
• Click on "Battery Replacement".
• Then click on "Bring in for Repair". You have to select a carrier, but it doesn't really matter.

https://www.apple.com/iphone-battery-and-performance/ - December 28, 2017 A Message to Our Customers about iPhone Batteries and Performance

iPhone Battery and Performance - https://support.apple.com/HT208387 - about power management, battery life

iPhone 6s Program for Unexpected Shutdown Issues - https://www.apple.com/support/iphone6s-unexpectedshutdown/- check your phone eligibility for a possible free replacement

otherwise:

iPhone Battery & Power Repair - https://support.apple.com/iphone/repair/battery-power - battery replacement and cost, plus reduced cost for some models

Genius Bar appointment - https://www.apple.com/retail/geniusbar/

How to find your nearest Apple Authorized Service Provider (AASP) or Apple Authorized Distributor (AAD) - https://locate.apple.com/country