Read your contracts, especially support staff. Depending on the school you work in, your contact will vary. In some schools under their contract, it will be support staff who do this. Just in the same way they administer meds, set up feed pumps, do physio and more depending on the school needs.
Generally teaching staff don't have this in their contracts. However it varies on the type of school.
2 metres concerns will be dismissed because it's a medical thing surely? Plus support staff enter that zone regularly, some will be constantly in the 2 metre zone as part of their normal working condition in the class.
School nurses, unless it's a SN school afaik they are assigned to the whole borough. I say they, it's usually a couple of nurses, and many aren't just assigned to schools but they are community nurses. Like many within NHS, they might have also been sent elsewhere.
Personally, it's like everything, I shall watch and see how this develops as more info gets released. IIRC, a part of schools opening after Lockdown 1, was to have testing in schools. Or maybe I dreamt it because here we still are.