"Presumably the most likely to be involved would be a legitimate flight, such as air ambulance, police or traffic?"
The Met Police and air ambulance Twitter feeds have already said they aren't involved except professionally.
The aircraft appears to have been dark blue. The crane isn't that far south of the river, at 51.484869,-0.128156 while the crash site is 51.482574,-0.127299 so the post-impact descent would have been almost due south, about 250m away. The wreckage path seems to lie on a N-S axis along the road with the bulk of the wreckage ending up near the railway bridge. Some wreckage is on the buildings on the south side of Wandsworth Road, so it evidently hit them as well before falling into the road outside.