It's hard to answer here. What happens in the US will be different than in another country, and parts availability is a continuing issue throughout the tech sector.
Based on past posts here and knowledge of what one must do to remove a dead battery (it's glued to the deck!), I doubt any Apple Store will do a battery service on a Retina Macbook Pro on-site. Removing the old battery requires toxic, flammable solvents that cannot be stored or used in a retail setting due to OSHA regs, HAZMAT precautions, and local fire codes.
Contact Apple Support and ask about a ship-in service. It will likely be fastest. Taking it to a store will only mean extra time because the store will likely have to send your computer to the same service facility as you will.
Mac Repair - Official Apple Support
I had a positive experience with a ship-in battery service on an iPad. Battery service on Notebooks is different than iPads but I did a ship-in with my iPad 5 during COVID. I set up the repair online, and dropped the iPad at a UPS Store on a Friday morning. UPS did the packing and shipping. By the time I retuned home, (maybe 10 minutes later) I had an acknowledgment from Apple that they had been notified by UPS of its receipt.
The Apple service center in California received it on Monday morning and shipped the replacement out the same day. I had a like-new working iPad in hand before lunch on Wednesday. Not bad considering there were two weekend days involved. That was in the US.
Others who had a Retina Macbook Pro battery service for a Retina Macbook Pro say that Apple replaced the entire upper deck with a refurb deck fitted with a new battery, keyboard, and trackpad. If they use an available refurbed assembly, that should be a fast turnaround.
If you have that deck exchange done by an independent provider, it will cost you US$500-600 based on tear-stained posts here. Apple charges US$200 regardless of which method they use. I've also seen posts that independents are having a hard time getting the deck assemblies given the many variants in models and keyboard languages.
Bottom line from a fellow user: Based on many posts here I doubt you can get an overnight service. Gone are the days of batteries attached by screws that you could change in 15 minutes.