Hello Steven,
Contact the London Metropolitan Archives (LMA) as a first step. Their website can be found at:
http://tinyurl.com/8qrpuejThe “Education in London” link for the archives can be found at:
http://tinyurl.com/k8erwwhEven if the LMA do not have the information you are looking for, you may be able to learn where the records are housed for the particular school or schools you are interested in.
For Dorset, see the National Archives page for the Dorset History Centre at:
http://discovery.nationalarchives.gov.uk/details/a?_ref=31The subscription website Find My Past has a collection called “National School Admission Registers & Log Books, 1870-1914.” See the link at:
http://tinyurl.com/n74wcljCheers,
Dave Boylan