If a midwife comes across a pregnant woman who has been cut and has a daughter this should be passed to social services.
Health visitors routinely ask if women have been cut (if the woman is from a country where this is practised, which is more than you would think). The same happens - it would be passed to the local social work duty team.
That may be shared with police if there is concern a criminal offence may have or be likely to take place.
If girls are routinely "checked" I'm not sure how that would work. (and I didn't see the programme so not sure what the suggestion was). The age at which this is performed is different in different cultures/ countries so this could include calling in all 12 year old girls from a certain country and asking them to be examined.
The mandatory reporting that's in place doesn't really work because health professionals are reluctant to do it.
I would be very reluctant to tell a family to take their child to be examined intimately for no other reason than their country of origin.
If a family refuses would they be arrested, or their child taken into care?
It's one of those dilemas of a liberal society and we have to work to get the balance right.
A big factor is that it is not spoken about. The women I've spoken to (from different countries) say It happens and then is not talked about ever again. Even though the practice has gone on for thousands of years it is still taboo. It needs to be spoken about more, in a way that does not present it as a practice confined to a few "tribes" in the middle of nowhere. It is not, it crosses class and religions.
I have heard of mosques where it is spoken against but not churches.
There's a move to a "harm reduction" model in Egypt (and possibly elsewhere) where wealthier families take their daughters to a medical professional and the procedure is done in clean. safe environment. (Egypt traditionally practices more extreme forms of FGM)
If the procedure is clean and "safe" the argument health against it does not work. But it is a feminist issue not a health one.
Often the arguments against these type of issues will focus on the practical rather than examine underpinning idea. It's similar to the focus on toilets in the trans debate. Someone finds a practical solution and hey ho problem solved - but it's not.
Alice Walker wrote a novel about this "Possessing the Secret of Joy". I'm not sure if its in that one or somewhere else but she writes in a very angry way about prominent black politicians who never spoke out against FGM, such as Mandela.