Surely the Liverpool to Sheffield bit should be covered by TPE drivers, in much the same way as FNW Holyhead drivers used to drive the intercity trains on behalf of Virgin between Holyhead and Crewe until the early 00s. Only TPE Liverpool drivers need learn the traction and route and overall id hazard a guess that it'd be cheaper than having Nottingham drivers having to do the very early /late stuff.
However a small issue, TPE guards also, hence TPE to run service, because the EMR guards would likely be hidden ie Nottingham based guards especially if overnight stay Liverpool. More than 12 hours. Don't forget Since BR split up around 1994 a lot of top link guards at quite few locations at the time become intercity west coast guards working those services only.
Hence Holyhead for Intercity West coast and Virgin etc becoming and retaining a Guards depot only so no issue there work one Holyhead to Crewe, then pass back to Holyhead asleep in the FNW train, nice for them at the time. Ie the guards. Easy life. One leg journey.
However with the Drivers it was contracted out like other services were elsewhere.
Ie a FNW Driver would work a Intercity one to Crewe.
Then would work a FNW service back to Holyhead. Unlike the guard who could relax and pass back. However at Liverpool it wouldn't work TPE Driver only. Either TPE guards and drivers or EMR guards and drivers as well based at Liverpool. in reality. But in hindsight a few other locations have small depots, so it could work out well for EMR to have one at Liverpool for drivers and guards.