Can I suggest you contact the school re breakfast and afterschool club? I am not talking about the fun clubs (eg football, craft club) but some school run after school child care clubs, or they are run by someone else on their premises, for wrap around care.
EG our school runs a breakfast club from 7:45 and 2 afternoons a week it runs a club til 5 which is primarily for wrap round care and almost never cancelled. And then there are fun clubs which change and don't run every week etc, and are usually until 4:20/4:30, and may be cancelled at short notice.
We have lots of childminders who do drop off and pick up. I think if asked out school would refer you to one, or maybe to PTA who could give you some names etc.
BUT you need to understand that school day times are set in stone and have to be kept to. So our reception children should be in playground at 8:35, and parents must stay until they go in to building at 8:40. You cannot drop and leave until KS2. For pick up you have to be there at 3:20, or arrange for someone else to be.
If you found another mum to do it, you would need to think about how much to pay her, and what you would do if her child was ill.
FWIW my cousin used to drive a school run taxi (special needs), there were 3 children accompanied by an adult, he wasn't expected to be responsible for them.