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My apologies for the delayed reply but I have only just seen your post.  Thank you so much.  Yes that Mr R Waugh would have been my dad, Robert.

Thank you both.  That is very helpful.

My apologies, Comberton.  I should have realised that the the links you provided were both to multiple page docs.  These are incredibly useful and have now helped to answer several questions and close gaps.  Thank you very, very much for this.

Thanks for the responses.  I have never found a marriage certificate for Thomas and Emmeline/Emma.  I found this on FamilySearch - https://www.familysearch.org/en/tree/person/details/LBN2-DBN .  There is a reference to another child, Lily Marion Ellis, who I believe was born to Emmeline's second husband James Ellis, not to Thomas. 

The information from the various responses above, I believe, allows me to rule out the Lawhitton Heard connection and thereby the emigration.  Incidentally I found a newspaper reference in South Dakota to Thomas Danning in 1896, suggesting that the Thomas Danning who emigrated to New York in 1902 is not the same one who appeared on the South Dakota 1905 census.  I am focussing on the Liskeard area now, as James (son of Thomas) and Emmeline (his mother) were certainly there in 1901.

Thank you for that service record.  Can I please have a link to that to study it more?

Thank you.  The marriage on 11 Jan 1856 was between my great great grandparents, John Waugh and Agnes Thomson.  She died not long after giving birth to my great grandfather, John Waugh - he is the one whose year of birth is given as circa 1856 on the census (he was actually born 9 March 1857).  Agnes' parents were John Thomson and Mary Menzies. 

Mary Hastings' son John Waugh remarried, to Agnes McKie, in Nov 1862 and had four more children (Catherine Howat Waugh, Mary Hastings Waugh (not to be confused with Mary Hastings - this is the granddaughter of the Mary Hastings I am researching, Jane McKie Waugh (later Pirie) and Helen McKie Waugh. 

Andrew also got married, only a few months after the 1861 census, to Jane Gordon, and they had six children.  This suggests Mary lived alone from late 1861 until her death.  She is not on the 1871 census, as far as I can tell, suggesting she died some time before then, although she could equally have moved away by then.

Mary's son John (b 1836) died in 1872 and her grandson John (b 1857) married Margaret Bell in 1881 and they had four children, the eldest being called (yes, you guessed it) John (1882-1942, my own grandfather)

The census information causes some confusion.  Both the 1851 and 1861 census list Mary Waugh as a widow, born c 1818, living with sons John and Andrew in King Street, Troqueer but the details differ slightly.  The earlier one gives a different place of birth for her and slightly different years of birth for the boys.  The earlier one also listens an extra child, Sarah McKie, listed as Mary's two year old daughter (if this is the case, she would have given birth aged 31, and possibly from an extra marital relationship given the differing surnames).  I can find no trace of Sarah beyond 1851, so it is unclear whether she died young or was possibly "farmed out" (sent to live elsewhere with another family), which was known to happen to children of poor families in that era and in that region as I know it did happen with other ancestors, but cannot prove in Sarah's case. 

It is possible, but seems unlikely, that there could have been two different Mary Waughs with sons of the same names and similar ages living in the same street ten years apart.  Discrepancies between the censuses may well be recording errors.

The 1841 census is an interesting one.  1882 John did move from Maxwelltown/Troqueer to Glasgow sometime around 1900 but I had never considered before that ancestors several generations earlier could have originated from there, so will look into that.  Thank you.