Hey!
I have Screen Time on my Iphone (to stop myself from going on it so much haha), so I tried to sign out of my Apple ID, but there was a text saying Sign out is not available due to restrictions. Then, I disabled my screen time and it was letting me sign out with no problem. From that, I believe that screen time will stop your child from signing out of their Apple ID.
I don't think the screen time restriction will be active if they successfully sign out and create a new Apple ID: it just wouldn't make sense honestly. Then again, like I said, I don't think they will be able to sign out since it stopped me.
My screen time is set up on my device and is not apart of the Family Screen Time Manager (I don't know the name, my phone isn't in English, sorry!), I don't know if that changes anything nor how your screen time is set up, but I thought I'd mention it just to be clear.
Hope I have helped you!