My interest was piqued even further when I read the following on a website devoted to the village:
In the mid 19th Century The Vicarage was extended to become a boarding school and boasts among its pupils William Gladstone who later became the Liberal Prime Minister.
I was not so sure. The biographies all agree that Gladstone was first educated at Seaforth, before being sent to Eton at the age of 11.
Then I came across Monica Raynes book Geddington As It Was and the mystery was solved. She quotes A. C. Ainger's Eton 60 Years Ago from 1917 on the school at Geddington Vicarage (Ainger was himself a pupil):
The school undoubtedly stood high in popular favour; it was always full and there was a strong flavour of aristocracy about it ... Nearly all the boys went to Eton ... Among the pupils were one cabinet minister, Lord Gladstone (son of the Mr Gladstone); two bishops, Augustus Legge of Lichfield and Edward Talbot of Winchester; three admirals, Charles Scott, Walter Kerr and Arthur Moore; and two generals, Ralph Kerr and Neville Lyttleton.
William Henry Gladstone was the Grand Old Man's eldest son and himself an MP for 20 years. He sat for Chester, Whitby and East Worcestershire. He also played football for Scotland in an unofficial international against England.
And Neville Lyttelton (not Lyttleton) was the father in law of my favourite Edwardian Liberal Charles Masterman.
Monica Raynes further quotes Ainger on the Geddington school:
There were only twenty pupils, so that personal supervision was easy enough; we were well fed, lodged and looked after - I believe that in the twenty years or so of the school's existence no inmate of it died.
The school was owned by the Revd William Montagu Higginson Church. In his The Rise of the English Prep School, Donald Leinster-Mackay writes:
Perhaps the reason why the school kept by the Rev. William Montagu Church at Geddington, near Kettering in Northamptonshire, was shortlived was that he was unable to control his temper, despite the school's being patronized by the sons of nobility.