Robert Gillingham's service records have not survived and it seems likely that he died in service, probably in Singapore or the Malay states between 1874 and 1878.
Son Alfred was admitted to the Royal Military Asylum in 1879 aged 11 years 2 months (so around March or April 1879). Parents named as Robert and Kate.
The 1861 and 1871 British Army indexes (compiled from muster books and paylists) have the following entries for soldiers named Robert Gillingham in the 10th Regiment of Foot:
1861: Corporal, No: 274, 1 Battalion at Aldershot
1871: Colour Sergeant, No: 1740, 2 Battalion, Rangoon
You can trace the movements of each battalion month by month via "Stations of the British Army" in contemporary newspapers, the Naval and Military Gazette being the most informative.