I'm just relying on my general knowledge of Victorian Britain (read a lot of Anthony Trollope) to make an edumakated guess that the fellow in the white trousers is a supervisor: he's got a full beard while working class men tended to be clean shaven; he's wearing a frock coat and a bowler hat, both marks of the middle class (note that the guys doing the actual shovelling are wearing cloth caps or wool hats; the upper class wore top hats); and he's wearing white trousers (which were fashionable at the time and are hard to keep clean). He's probably the contractor or engineer in charge of the demolition.
I'm not as educated on the niceties of dress as a Victorian would be, but they say that you could tell the exact social position of a man or woman by their dress, hair, hat, accent, etc. They were more concerned about social class than modern Brits, and a LOT more concerned about it than any American now or then.
St. Giles is just outside the old Wall of the City -- it appears from maps of the fire that the old wall acted as a firebreak to some extent, since the fire follows the outline of the Wall. Or that may just be where the city fathers decided to demolish some houses to stop the progress of the fire.
One of the surprises after the Blitz (it's an ill wind that blows no good) was all the Roman and even prehistoric remains that turned up during the rebuilding. A huge piece of the old Wall is preserved right behind St. Giles -
You should recognize the really ugly building in the background as it's also in the background of the photo of St. Giles - part of the modern Barbican development.
P.S. . . . when I was referring to "the XXXIX" I meant the Thirty-Nine Articles of Religion. They've been relegated to the status of "Historical Documents" in the back of the Book of Common Prayer, but back in the old days if you got ordained or even if you became a student at one of the universities you had to affirm the XXXIX. They are a bit on the 'low church' end of things, and most Anglo-Catholics simply ignored them, especially Articles XXII, XXV, and XXVIII.