OP - this is a funded scheme in my area and had been for ages. Is it not in yours? That's how the brushes and paste get paid for in DS2's nursery and now DS1's primary school as well. When DS1 was in nursery, we had to send him in with a toothbrush, but they're now provided as part of the scheme to both nurseries and schools.
It terms of storage, our nursery has this tray thing with a big cover over it that the toothbrushes stand up in so they're kept quite separate but still altogether - it just sits up on a shelf when not in use. For extra hygiene, each brush has a little brush cover as well.
As I understand it, the school and nursery staff don't brush the children's teeth for them - it's supervised brushing. DS2 is way too young (3.5yo) to brush his teeth completely independently yet, so I'm really pleased he's doing this at nursery as a practice, then gets two proper brushes at home morning (yes, always before breakfast, not after) and evening. So in terms of staffing, one staff member could easily supervise four toothbrushers at a time - one at each sink, perhaps - and keep to your ratios. I see it more of an independent life-skills practice, rather than full responsibility falling to nurseries for maintaining the dental hygiene of the nation's under-4s.
So, in terms of your nursery workload I completely understand where you're coming from - nursery staff do a bloody amazing job on shit pay and this seems like yet another thing you're being expected to do with no remittance. However, I think the support for the scheme probably is out there (PPs have mentioned links above) plus I think if you view it as part of the EYFS self-care goals, rather than rotting teeth being 'your fault' it might make it a bit easier to bear perhaps?