I've just spent around 10 minutes trying to find an answer to your question on my local hospital's website, without success. The Government's press release doesn't help either:
https://www.gov.uk/government/news/face-masks-and-coverings-to-be-worn-by-all-nhs-hospital-staff-and-visitors
I suspect that some common sense is needed here. i.e. if you've got a 9yo or 10yo child who's perfectly happy to wear a face covering, then it would be best to get him/her to do so. If you've got a 2yo who'd be terrified of having their face covered, then don't bother.
Although they'd probably prefer people to arrive with their own face coverings, hospitals will have them available for people who turn up without them anyway. (Indeed, for a forthcoming appointment, I've been told to knock on the door of the clinic and I'll be given a face mask. There was no mention at all of having to provide my own). So simply play it by ear and see what happens.