If the original verses do indeed date back to 1706, there would have been no Scottish Regiments in the British Army

"Were there any Irish regiments stationed in Ceylon between 1795 and 1860?"

The only possible candidate is an "Irish Regiment" the Leinster Regiment whose first battalion originated and was originally raised in Canada and its Second battalion that was part of the East India Company's Army.

The only fighting was early on in the period you mention the Kandyan Wars all over by about 1819 if memory serves the British Regiments who fought then were the 51st Foot and the Green Howards.

None of which, goes to say that Irish soldiers were not present in Ceylon, by 1866 (I know it is outside your time frame) two-thirds of those serving in British Army were Irishmen.