Professional transport planners use the concept of O&D Prediction - Origin & Destination, to analyse travel flows. This must surely have been done, in great detail, for Crossrail, and yet the figures, which would include the diversion of flows from established means, whether NR, other TfL, road, etc, have never been published. I suspect that they must be so concerned about being embarrassed if things turn out differently that they would rather keep all this, on which billions of public expenditure depended, to themselves.
Possibly the only two key players who would have understood these figures are Sir Peter Hendy, onetime Commissioner, and Andy Byford, current commissioner. Of the intervening Commissioner, and the rest of the senior TfL/Mayor hierarchy, the less said the better. Obviously figures and flows have changed since Covid, but Sir Peter's onetime comment that he feared "Crossrail might be full up from Day 1" seems a notably informed comment.
Regarding flow change, I do expect it will draw passengers away from the four east-west Underground routes across central London (Jubilee, District, Central and Circle North Side) in varying proportions. Jubilee quite substantially, as it offers a whole range of shortcuts right out to Stratford. Central likewise, being the closest. However, there will be some diversion TO these lines as well. Kings Cross to Canary Wharf, presently via London Bridge, moving to change at Farringdon, for example.
The DLR I anticipate may end up losing traffic. Beckton/Excel to Central London, either direct or changing to the Jubilee at Canning Town, moving substantially to the Custom House change. Canary Wharf to The City likewise.
The Piccadilly may be among the greatest to benefit, with its Heathrow traffic from central London moving over very substantially. The present day routing via Paddington is tedious and offers no advantage to get to the airport, hence why the Piccadilly retained so much of the traffic, but Crossrail is going to make a wholesale change in flows there.
There are also some secondary effects. Routes which are currently overloaded and have passengers diverting away will come back into into convenience. Liverpool Street westbound on the Central, or Canning Town westbound on the Jubilee, both of which suffer from major inability to even get in the train in the morning peak, may come back into favour for many.