If I understand correctly, you are using the Apple Watch Family Plan, where the dependent children Apple Watches are controlled by your (or your spouse's) iPhone.
You are getting one of the children their own iPhone (it MUST be an iPhone 6s or newer), and you would like to remove that child from the Family Plan and then pair that child's Apple Watch to their new iPhone.
Yes you can do that.
Details on cancelling the Family Plan cellular line, and getting a cellular contract with a cellular carrier that supports the iPhone and Apple Watch are details to be worked out with the cellular carrier (or carriers if you are switching to a different carrier for the child). It should also be possible to transfer the cellular phone number from the Family Plan to the new iPhone, so the child will not have to give out a new cell phone number.
Once the Apple Watch is paired to the iPhone, and the Apple Watch added to the iPhone's cellular plan, the Apple Watch will be able to access the Internet and make phone calls without the child's iPhone being nearby.
T-Mobile is what my Wife uses for her Apple Watch. Since she does not use much data on her Apple Watch, as mostly the iPhone is nearby or she is on WiFi, she even got a $5 per month add-on plan for her Apple Watch. Otherwise, T-Mobile charges $10 per month to add the Apple Watch to a plan (I do believe you need to have automatic credit card payment to get the $10 per month or the $5 per month options).
