The times I normally come across this happening is usually when the NRP has a proven history of domestic violence and the RP is trying to go about their lives unhindered obtain medical treatment, register with schools,de register from school that sort of thing.
With a state school the expectation of consultation is not down to the school it's down to either parent and the school or LA may ask for both parents signatures but they cannot insist on it,
How this tends to play out is quite silly but it happens lots will use de registering as an example (for home education)
RP who wishes to HE send de reg letter citing the section of the education act. Removes child from school (perfectly legal)
NRP waits until contact is on a school day and registers child (once child is registered its a legal requirement to attend) leaves child there.
This can be repeated loads as long as the school still has a place for registration.
It only stops when either party obtains a prohibitive steps order
RP would ask for an order forbidding him from registering NRP would ask for Forbidding HE.
I've also seen this happen with parents both registering at different schools its odd how weird people can be
medical stuff is the same nobody ever orders you to obtain both parents consent everytime I have known of one parent wanting it but the other not its been down to the objector to try and prevent it (unless hospital trust goes to court to force it)
Off the top of my head the only thing that at time of delivery (or attempt) where they can check or force you to evidence the other parents consent is leaving the uk (without a residency order).
If it was any other way round how on earth would the thousands of lone parents who don't even know how to contact the ex deal with anything?