Hi Daisy,
I'm back from my overseas trip including visits to the Family Research Centre in Islington, the National Archives in Kew Gardens and Ballincollig in Ireland. I found lots of very useful information which has unravelled the puzzle considerably.
I found my Great Grandfather Arthur Harding's Army Discharge Papers and also references to him in the Pay and Muster Books in the National Archives. These confirm that:
1. He was born in Croydon in April or May 1835;
2. He joined the Royal Artillery on 25 August 1857, serving in the 2nd battalion, 8th Brigade as a "Gunner/Driver" on enlistment;
3. He was promoted to "Collar Maker" on 1 February 1862 and was posted with the 8th Brigade to Ireland from about 1863 to 1866;
4. He was promoted to Corporal on 1 February 1867 in Woolwich and re-enlisted for a further ten years on 10 October 1867;
5. He was promoted to Sergeant Collar Maker on 2 October 1869 and transferred to 3 Brigade on 1 July 1877;
6. He was discharged on 2 October 1878 in Rawul Pindi, East Indies, after serving a total of 21 years and 39 days in the Royal Artillery, including several years in the East Indies (the actual figure has been partially scratched out and is hard to decipher but it looks like 10 (?) years and 294 days).
However. I have also located a marriage record for "Arthur Coventry Harding" - December Quarter 1871, in Richmond, Surrey and on the same record page is shown "Ellen Brooks".
An Ellen Brooks was born in Macclesfield, Cheshire, in 1845 - this is two years out for the Ellen Harding shown in the 1901 census, but given the birthplace it seems a very likely possibility that this is the Ellen who married Arthur Harding. The only difficulty is matching the length of Arthur's service in the East Indies with the marriage date of 1871, but on the other hand it does fit with John Harding's birth to Arthur and Ellen Harding in the East Indies in 1873!
While I was in Ballincollig I was able to obtain the Baptismal Certificates for my Grandfather and his sister, which confirm that their parents were Arthur Harding and Margaret Long. Margaret was clearly from a Catholic family, hence their baptism in the RC church. I also believe I have located her Baptismal Certificate from the same RC church in Ballincollig. She appears to have been born and baptised on 5 April 1840. Her parents were John Long and Judith Brian. Now here's a fascinating fact - the RC church also found a birth/baptism entry for another Margaret Long, born to the SAME parents, but 5 years earlier! The godparents recorded on that certificate are different (though there is a surname connection), so this is not a duplicated entry with a different date. It seems highly likely that the first baby Margaret died and the second baby girl (my Great-Grandmother and Arthur's wife) was named after her. Apparently this was a fairly common practice in those days.
What happened after my Grandfather was born in 1866 I don't know - but Arthur Harding senior clearly returned to England in late 1866 and re-enlisted in 1867.
I think it is possible that Arthur senior left my grandfather and his sister Eliza with their mother or her family when he returned with the Army to England and I am guessing that perhaps Margaret Harding nee Long died sometime after that. (I doubt that the marriage would have broken up because, being Catholic ,Margaret would not have agreed to a divorce which would have precluded Arthur marrying Ellen in 1871, if indeed that is what actually happened).
This certainly seems the most likely scenario at the moment and has opened up several more possible leads.
There is lots more I could tell you - this was a VERY rewarding trip for me! - but this post is already rather lengthy! Thanks for the leads that you provided, which helped me to match all this info.
Cheers,
Mike