You probably wouldn't use escalators - if you are going for a truly deep station you would probably just buy those
ThyssenKrupp linear motor lifts. Pairs of lift shafts, each with a continuous loop of lift vehicles.
Metro stations abroad are already trending towards simply throwing more lifts at problems as opposed to "traditional" escalator solutions.
Well if we went for a Barcelona style solution of two track running tunnels with stacked platforms in the tunnel bore, the actual "cavern" volume would probably be quite small.
And of course, once a tunnel arrives on site you can put a conveyer into it to remove spoil.
It would obviously be enormously expensive, but I don't think the project is infeasible in engineering terms.
I'm fairly certain it would still win for any journeys that it allows to be done with one change instead of two (onto the Underground and off again).