"Locking out a diversity of talents and experiences makes Britain's leading institutions less informed, less representative and ultimately less credible than they should be," warned Mr Milburn in his foreword to the report.
"This risks narrowing the conduct of public life to a small few who are very familiar with each other but far less familiar with the day-to-day challenges facing ordinary people in the country.
"That is not a recipe for a healthy democratic society."
Some 75% of senior judges, 59% of the Cabinet, 57% of permanent secretaries, 50% of diplomats, 47% of newspaper columnists, 38% of the House of Lords, 33% of the shadow cabinet and 24% of MPs hold Oxbridge degrees.
In contrast, less than 1% of the whole population are Oxbridge graduates while 62% did not attend university, says the study.
The report describes the figures as "elitism so stark that it could be called social engineering".