A bit pedantic, but "the bit between Liverpool Street and Shenfield" is a bit false because that is simply the existing metro service that's run on that line for decades, the only relation it has to Crossrail is the use of the Class 345 rolling stock. Since they still only run half-and-half between those and 315s (though more off-peak), I wouldn't consider even that complete either. There are good reasons why the new fleet hasn't taken over yet, but nonetheless, to state Crossrail is already running on that line is basically meaningless.
Hwl's estimates are the current thinking. If I were to offer my conjecture I would hazard a guess at Autumn 2021 for Phase 3, Spring 2022 for Phase 4 and Bond Street, Christmas 2022 for Phase 5 and 24tph core operation.