How to renew AppleCare for my Apple Watch?

Are you referring to AppleCare (the default, 1 year warranty) or AppleCare+ (the longer, subscription-based protection program)?

If you didn't buy AppleCare+, you can't extend it now. AppleCare (the normal warranty) cannot be extended.

If you do mean AppleCare+, see the link below. The guidelines are different for every device and country.

Extend your AppleCare+ coverage - Apple Support

The original one year warranty is this (not referred to as Applecare):

Apple One (1) Year Limited Warranty

There is still an extended Applecare plan available in a few countries and you have 365 days to purchase it from date of purchase of device.

The Applecare + is a more recent addition to the lineup and needs to be purchased within 60 days.

I thought this used to name the countries, but no more (at the end - last paragraph):

Buy an AppleCare plan - Apple Support

Ah, I see. Thanks!