Seeing as Jack Joshua was a butcher, with several sons, I wonder if your family might have been the Collins Bros who ran a butchers by that name on Chase Side, Southgate? It was to the left of the Crown Hotel, near Chase Side's junction with Crown Lane. It disappeared in 1963 when both premises were redeveloped into shops with offices above.
I never knew it personally but have the local history book "Southgate, A Glimpse into the Past" (1987) by Alan Dumayne and with so many Collins on your list, it seemed worth a check of its index! The book has a photo of it too.
You might be able to get more info via the local history staff for the modern day borough of Enfield -
www.enfield.gov.uk then select "Leisure and Culture" from the list on the left, then "Local history and heritage". They might be able to check street or telephone directories, rate payers records or the electoral roll (the butchers had an upper floor so someone may have lived above the shop). I know they have the same book there.
Good luck, Sharon