I did see the Auchtemutty Mr Kane who transferred to the 59th Battalion. Alas, like you pointed out, this chap was not the right age for your man.
With reference to this part of your post.
The next mystery is that during 1857/58 the 59th were stationed in Hong Kong/China, and had been for several years. So how did he come to be born in Ireland? If his mother didn't travel with them, how was he conceived? If she did travel, under what circumstances might she have been sent to Athlone? Although the Regiment had its Depot in Ireland, I can't find any evidence that it was at Athlone. For it to have been recorded as a military birth, I'm assuming he was born within the barracks.
Is it possible James Kane / Keane may have been with the 59th Depot battalion?
If so, they were responsible for recruitment, training, sending of reinforcements. The Depot of the 59th was in Ireland right through the 1850s.
From
June 1850 Fermoy
Jul 1850 Tralee
November 1851 Fermoy
November 1852 Charles Fort
September 1853 Kinsale
October 1853 Charles Fortt
February 1854 Kinsale
October 1854 Cork
July 1855 Belfast
December 1855 Londonderry
August 1856 Curragh
September 1856 Athlone
November 1860 Chichester.
It may explain how Thomas may have come to be born in Ireland?
I found a James Kane and a James Keens of the 59th but they were in China.
Sorry I couldn't be of more help. I wish you well in your future investigations.
You only live once, but if you do it right, once is enough.
Best regards,
Dave