See this link to schedule battery service. Be advised Apple doesn't actually replace the battery in Apple Watch. For a fixed price, they take your watch, and in return give you a refurbished watch of the same model and color.
Enter your information here --> Apple Watch Service & Repair - Apple Support
I know you say the watch still meets your needs, but it is now quite old and cannot run the latest WatchOS versions. You may want to give serious consideration to applying the money you'll spend on a battery service towards the purchase of a newer, more modern watch. Your choice, of course.