They will give you a refurb unit.

Probably you will get a refurbished Apple Watch with a new battery

They charge more the older the model is. I just did this, I don't have Apple Care, and it cost me around 20,000 yen (I'm in Japan). Don't be surprised if it costs you over $100.

You do get a whole new refurbished unit though, which is nice.

You take it in, they check it for other damage besides the battery and if there is none then they accept it for battery service.

You do get a refurb, but this is an Apple refurb so while the internals may have been used before, the case will be completely new and I distinguishable from an off the shelf item.

US standard is $0 if you have AppleCare+ and $79 if you don’t, across the full range, but only as far back as the S2.

Add an extra $10 if you want the replacement shipped back to, it’s free to collect it at an Apple Store.