Crossrail 2 should be cost-effective, so what about taking over Thameslink (between the Sutton loop and Luton) services, the Moorgate to Finsbury section, the route to Welwyn Garden City and the Dagenham Docks branch (Tower Hill to Grays via Dagenham Docks)?
That way, it requires little construction (apart from tunnels between Moorgate and between F.S. and Limehouse) and not much need for new trains (simply take 700s, 717s and cascade 707s from SWR), thus allowing a cheap plan for Crossrail.
There's no need for new tunnels somewhere in the 2020s as there's plenty more to take over (like that Moorgate line).