Not sure how old you are and where in the country you were educated but plenty of people in their late 40s and older educated in inner city areas remember their primary school being involved in their dental hygiene.
In my case it was a mixture of my teachers giving me lessons before I was 8, then dental check ups in school until I was 10.
My DD is was taught dental hygiene by her childminder and nursery separately. They both gave her a toothbrush and toothpaste, as well as sending her home with a leaflet about toothbrushing to me. In her primary school, like it has been for the last few years, they are very prescriptive on snacks and packed lunches. In fact as we are in London they discourage children from bringing in packed lunches.
I've heard already in some deprived areas they already get TAs to do tooth brushing with children.