how do I update my watch series 6 when I am not paired to an iPhone.

It's impossible for you to not have the watch paired to an iPhone. By definition an Apple watch cannot function without an iPhone so an iPhone must have been used to set it up either directly or through Family Setup.

In either case you must use the iPhone that was used to set it up to complete the update. There is no way around. this as Apple watches cannot be setup without an iPhone period.

Hello,

If you would like to update your Apple Watch just from the watch follow these steps:

Make sure that your watch is connected to the internet.

Go to the Settings app on your watch.

Go to General > Software Update

If you see an update available to you then tap install. And follow given instruction on screen to you.

Leave your Apple Watch on charge while updating, make sure that you DO NOT restart your Apple Watch while updating as it'll void the update, the Apple Watch will restart on its own after the update has completed.

To my memory I do not know any other way to solve this. On your Apple Watch is there no where for you to find the T&Cs?