This is a horribly messy situation and could end up taking several long replies to resolve.
Apple's mobile devices can only be associated with a single account (for most purposes) at any one time. Ideally the child would abandon use of one account and both the child and your wife would be members of you family sharing group. If your child is willing to abandon account #2 that is simplest because then your wife can reset the Watch with no account, request deletion of child account #2 account, then delete her family sharing group, then both your wife and child join your family group, set up the Watch again with account #1.
If the above does not appear to be a satisfactory solution then let us know the issues and we can maybe resolve them.
Deleting A Child’s Apple Account
One way to do this would be to log into the unwanted "child" account and request deletion of the Apple ID. You should have the ID and password from when you first set up the account.
Another way would be for the child to make the request.
From: How to delete your Apple ID account - How to delete your Apple Account - Apple Support
Section: Can I delete my child’s account?
"Your child can also request deletion of their Apple ID, which requires your approval.
The approval request will be sent via email to the primary Apple ID email address of your family organizer account. When approving the request from this email, you will go through the process of deleting the child account."
Disband a family group --> How to leave or remove a member from a Family Sharing group - Apple Support
What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support
If you want your wife to be the Family Sharing Organizer you could instead transfer your child 1 account to her family group as instructed in How to transfer a child to another Family Sharing group - Apple Support. Then you can disband your group and join hers too.
Another take on this would be to keep both child accounts (but only one can be on the Watch) so that any purchases made with the abandoned account can still be shared with the new account. Note that
See what I meant by this can be messy? ;-) There's a lot of ways of dealing with this and the best way will depend upon what exactly you want or are willing to accept in the end.