You are almost there...

Just that it would not need any digging at the LHR end - all the tunnels are there already and in daily use.

Basically an upturned "Y" with T5 and T4 at each point and T2, T3 in the middle

It is a commercial decision to send which trains to which terminal

It would be perfectly possible to route HEX trains to T4, Heathrow Connect Trains to T5 and vice versa - it will also be possible to route the Crossrail trains to both too... (as I suspect will eventually happen)