First, the thing on the end of the cable I described as about six feet long is actually 20ft or more - it turns out to be a VERY large hydraulic grab with two buckets that close like jaws. Couldn't get a picture, but it turns out to have been in one of the ones I took at Royal Oak Station - it's the thing to the left of centre.
A couple of months ago they put in a steel retaining wall between the crossrail works and the Hammersmith and City line, which I assume is temporary. What I think they're doing now is cutting a deep slot behind it, say 4ft wide and 20 or 30 ft. deep, several hundred yards long, and filling it in with concrete as a permanent barrier. There's another machine that looks more like a conventional mechanical digger (but has the same massive hydraulic thing) which I assume does the first pass, say down ten feet, then they drop this thing down the slot to excavate it to its final depth. That's in the centre of the picture above, with the cab etc. completely hidden by the jaw thing.
I was surprised by the size of the hydraulics on both machines, they look enormous, then I realised that they are going into London clay that has had a couple of centuries of being pounded flat by endless rail traffic - it must be very dense and completely compacted.
Mystery solved, anyway.