No. You will need to cancel the plan on the watch you currently own and set up a plan on the new watch.
You also need to see if your Christmas gift can still be returned. Apple did extend returns for holiday items to January 8th, which you're past.
See this from Apple:
Items purchased at the Apple Online Store that are received between November 1, 2021 and December 25, 2021, may be returned through January 8, 2022. Please note that all other terms and conditions provided in the Apple Online Store Sales and Refunds Policy are still applicable with respect to such items purchased. All purchases made after December 25, 2021 are subject to the Standard Return Policy.