Moving my son's apple watch (with cellular) set up on family plan from my phone to his new iphone

My son's Apple watch was setup on my phone using Family Setup (It has a cellular plan with it's own phone # so he can make calls and text. He now has his own phone (which has a different #) and I'd like to cancel the watch cellular plan and pair it to his new phone. I see steps for pairing a watch to a new phone, but it doesn't mention anything about Family Setup. Are there any extra steps I should be taking if the phone is currently setup using Family Setup? He doesn't want to lose his contacts and I'm trying to avoid the need to manually reenter the contacts on his phone. Since he has his own Apple ID, would that automatically transfer his contacts/info to the new phone?

Also - I do want the new iphone to remain under my family plan which I assume is no problem since my other child's phone shows in family plan and I can set parental controls for both - I believe that has to do with their apple ID, and goes across all devices using their apple ID...?