Traveline is good for finding out about public transport services including the ones which do the school runs but are not council-provided contract buses. They have a website and you can e-mail and telephone too - they have always been ultra efficient when I have used them. They will also give you the numbers of all the bus companies so you can find out about fares etc. http://www.traveline.info/
It is also worth contacting the schools direct. Some schools run their own subsidised bus schemes and many can give you advice about which services/routes pupils use to get to school.