Hi
I've looked into this quite a bit.
Information about schools can first be obtained from the `Get information about schools' database, formerly known as Edubase (it can be downloaded as a giant csv file). This can give you school type, admissions policy and the percentage of students on Free School Meals. POLAR4 data on higher education participation, Indices of Multiple deprivation data, and ethnic group data are all tied to census areas (MSOAs and LSOAs respectively). It is possible to look up the school's resident MSOA/LSOA to get these metrics, however, school catchment areas are much larger than census areas. The London Schools Atlas allows you to find all the census regions within a schools' catchment area, but that information for the whole of the country is not publicly available and would require requesting access to the National Pupil Database.
Regards
Martin