How do I know when my battery needs replacement?

To understand Apple Watch performance and its relation to your battery, see:

Apple Watch - Battery - Apple

Apple Watch Battery and Performance - Apple Support

“Battery life” is the amount of time your device runs before it needs to be recharged. “Battery lifespan” is the amount of time your battery lasts until it needs to be replaced. Maximize both and you’ll get the most out of your Apple devices, no matter which ones you own. See the "Tips for Apple Watch" section in:

Batteries - Maximizing Performance - Apple

Also see:

When to charge your iPhone or iPad - Apple Community

Finally, keep in mind that batteries are "consumables". Their capacity starts dropping the day they come off the assembly line. Battery life is determined primarily by "full charge cycles" - A full charge cycle is 0% to 100%, or any combination that adds up to 100%, such as 50% to 100% or 30% to 80% twice, 80% to 100% 5 times, etc. The battery will gradually lose capacity, and is rated to remain above 80% capacity (that's total capacity, not charge level) for 500 full charge cycles for an iPhone (1,000 for an iPad). For most people this will be around 2 years for the iPhone. Apple will replace the battery for a reasonable fee (Use this link to find the cost for your model and country→iPhone Battery Replacement - Official Apple Support) if it goes below 80% after the original 1 year warranty or AppleCare+ warranty period. Within the warranty the replacement is free.

For repair & service:

Apple Watch Service & Repair - Apple Support

You don't say which watch you own, but you can check the health of the battery by opening the Settings app on your watch and scan down to Battery, then Battery Health. If Battery Health is near or below 80%, the useful life of the battery has been spent.

Whether you should invest any money in the watch or not really depends on the model and whether it is modern enough to run the newer WatchOS versions.

The watch is modern enough to run the latest WatchOS versions. Apple Watch 9 will be coming out this fall. Personally, I tend to upgrade after 2 years with Apple Watch, but that is me.