We know from Henry's son's christening record that Henry ARCHIBALD was "a travelling Scotchman" and that Henry's father was Robert ARCHIBALD, late, of Musselburgh, coalminer. Henry was born on 29 Mar 1766 and baptised in Newton, near Musselburgh, Midlothian.
It is possible that he left Scotland when some of the mines where his family had been working were closed due to drainage and geological reasons in 1790, or perhaps he ran away to escape the enforced bondage associated with coalmining in the 18th century.
The most plausible explanation is that he joined the 9th (East Norfolk) Regiment of Foot and was the Henry ARCHIBALD, born Dalkeith (2 miles from Newton where Henry was christened), who was pensioned out of the 9th Foot on 8 Jun 1790 aged 23. The 9th Foot were embarked to the West Indies in 1788, and it was on St Kitts at Brimstone Hill Fort - now a UNESCO World Heritage site - that he lost his right hand while 'blasting a rock'. His discharge papers give the date of his accident as 3 Dec 1789 and the surgeon officially signs his discharge papers on 15 Mar 1790, presumably once it was clear that he would survive and need to be discharged. The papers also give his age as 'about 22 years', height as [5?] feet 5 1/2 inches, dark eyes, complexion as 'swarthy', with 'a long visage' and 'dark hair'. His trade was recorded as 'miner' which aligns with his father's trade. His pension reflected his 6 years service, good record and he was "humbly recommended as a proper object of His Majesty's Royal Bounty of Chelsea Hospital".
Perhaps this experience led to his "travelling Scotchman" epithet.
Another researcher has suggested that Henry was married to Elizabeth LINDSAY and had two daughters, Agnes and Allison, before leaving Scotland, although I've not found evidence of this and it doesn't fit the compelling military narrative above.
Henry married Mary SAGAR in New Malton in November 1796. It is believed that their first child Robert was christened in September the following year in Manchester. That child died in infancy, and another Robert and a Henry were born and christened in Skipton in 1801 and 1804. It's possible that they were staying with Mary's uncle Edmund SAGAR who was a clockmaker in Skipton.
Henry died in Blackburn in 1832 and was interred at the Chapel Street Independent in Blackburn. This suggests that he was a non-conformist which would be consistent with his Scottish upbringing. His son Robert married another non-conformist, Harriet FIRTH.