Let me return to my workings on this.
Who's Who lists the late Mr Richard Cole-Hamilton's father as "John Cole-Hamilton CBE". The latter's Who's Who entry states he's the son of 'Col A R Cole-Hamilton'. John C-H's entry in the Companionage section of the 1973-74 Debrett's (p2690 col 1, the last edition to carry such a section) confirms he's the son of 'the late Col A R Cole-Hamilton of Beltrim, Co Tyrone".
Col Arthur Richard Cole-Hamilton's entry in the 1910 edn of Kelly's Handbook (p398 col 2) is as follows (note it doesn't list his children):
Cole-Hamilton, col. Arthur Richd., eld s of of capt Wm. Claud Cole-Hamilton, of Ballitore House, co. Kildare (d. 1882), and grands. of Maj. Arthur Willoughby Cole-Hamilton, of Beltrim, co. Tyrone (d. 1892 sic, Burke's gives the correct date of 16 Dec 1891); b 1859 m 1, 1882, Jane, eld dau (d 1883) of Samuel Moore; 2, 1884 Florence A., dau of late Jas. Duke Hughes... [address] Beltrim Castle, Newtown Stewart, co. Tyrone; Bellewstown House, Drogheda.