Hi CA
There is very little on the web about the Edmonton Rifles. There is a mention in Hansard that gives (in 1856) their full title as the Edmonton Royal Rifle Regiment of Militia. They were also known as the 2nd or Edmonton Royal Middlesex (1862 Army List) so I think the officer in charge in 1871 was using an old name to fill in the census. In 1862 their HQ was Barnet, which fits the 1871 Census. I cannot find them in the 1883 Army List, so they may have disappeared once the Militia came under the War Office.
Regarding his service, here is a link to researching the Militia in Kew -
http://www.nationalarchives.gov.uk/catalogue/RdLeaflet.asp?sLeafletID=26. It looks as if he was discharged (with a pension) from a regular regiment and then enlisted in the Militia, probably on the permanent staff. The problem is that you need to know his regular regiment in order to look for his papers for his first period of service. His Militia papers (WO96) should give his previous regiment. You can then look for him under that regiment.
Researching regulars -
http://www.nationalarchives.gov.uk/militaryhistory/army/step4.htmFailing that, I think his children are the key:
= Joseph Nicholls, 52, Staff Sergt (and) Chelsea Pensioner, born: Spitalfields, Middlesex
= Ellen, 33, born: Ireland
= Mary E, 13, born: Jersey
= Ellen C, 11, born: Shorncliffe, Kent (a well known garrison town)
= Elizabeth B, 6, born: Barnet
= Sarah A, 4, born: Barnet
= Martha J, 2, born: Barnet
= Louisa T, 5 months, born: Aldershot
I think you can discount the last four as these look to be while he was in the Militia. I could not find the birth of Ellen on FreeBMD, so you may have to visit Kew and look in the Army Lists and see which regiment was in Jersey abt 1857 and Shorncliffe (abt 1860).
There is a marriage on FreeBMD of a Joseph Nicholls in Sep (qtr) of 1856 in Lambeth, either to an Ellen Kirk or Ellen Crawley. If this is his marriage the certificate may give his regiment.
Ken