An enhanced DBS checks all the same things regardless of who asks for it, I don’t know if the school ones would ask for a barred from working with children check.
I’ve got one done online through the church and is exactly the same as my cadet one and took the grand total of 4 days; from start to certificate on the door mat and the person who is responsible for safeguarding got an email notification to say it was OK, none of this photocopying, signing and postage nonsense we have to go through. The only reason the Air Cadet ones take so long is to keep people in MoD jobs. I would imagine anyone working in schools, children or social services would be done the same and have ones that are acceptable.
There is a world out here where things happen without ridiculous hoops and delays, mainly because organisations and businesses don’t / can’t afford employ loads of people to shuffle paper or justify a highly paid and convoluted management structure, so they remove or reduce admin or make it work better for them.