Does the stimulus check get automatically applied to back child support like your income tax returns do?

Yes.  The stimulus payment can be intercepted for back child support owed.

How long does it take for that intercepted stimulus check to be applied to the back child support owed? 

TurboTax does not know how long it will take for an intercepted stimulus check to go to the ultimate recipient of the child support order.  That may depend on the agency that is enforcing the child support order and how they get it to the right parent.