I am in the same situation. I have the 42mm space grey sport and the stainless with classic buckle, with link bracelet on the way..May shipping.
You have to un-pair the watch that you are not using at the time and pair the one you are using. The easy thing is that you can restore from backup..just make sure the watches have two different names, as your iPhone saves multiple backups.
The only issue with this is your credit cards entered in apple pay will be removed when you un-pair. When you pair the watch that you are using, you will need to re-authenticate the cards with apple pay.