If Edward's father was a veterinary surgeon and served in the Crimean War, his name should appear in Hart's Army List for the war years (as it lists all officers who served during that year in the British Army). I don't have the earlier versions (ie. 1854-1856), but I have a copy of Hart's 1860.The only Keough or Keogh's listed include: Anthony Robert Keogh (p. 202)who was a lieutenant in the 14th (the Buckinghampshire) Regiment of Foot and did not see Crimean service; Henry Keogh (p. 156) who was a lieutenant with the Military Train and did not see Crimean service; Lynch John Keogh (p. 156) who also served as a lieutenant with the Military Train and did not see Crimean service; and William Keough (p. 197) who served as a lieutenant in the 12th (E. Suffolk) Regiment of Foot, also a non-Crimean vet. You might look at earlier versions of Hart. The simplest thing would be to pay a researcher . . . and Major Brian Oldham who does research has the most complete list of Crimean veterans in the world. You may find out more about him at http://www.wolstadt.demon.co.uk/http://www.wolstadt.demon.co.uk/
Needle in the haystack approaches include poring over the Cook's Casualty Roll for the Crimea or trying to find something via census records or on the Crimean Medal Rolls which are on CD but run to I would guess thousands of pages with no adequate search tool.