Simply the Worst

This route is 5:30 minutes with no WiFi or power and to make matters worse it is on the slowest possible route. Well equal slowest but you have to go via Lincoln to get to the same time

Just to given an idea of how appalling this route is I worked out all the alternatives which turn out to be something like.

or in the real world.

Note the lines in black on the map below, no longer exist but would make the journey quicker. The red bypass under Derby has no passenger services so all timings involve 5 minutes turn around in Derby, there was a bit of track to allow a turn east while passing through Derby north but that has gone.

                                                       

After messing around with the National Rail route planner and discovering it is very buggy and very limited and totally unfriendly I got these timings. These are for the fastest journeys between various points, you cannot make these journeys at this speed now because you'd have long gaps waiting for connections. 

 The UK rail system is now so messed up you cannot do a lot of the routes without multiple tickets. By a strange fluke, the slowest route I could find has exactly the same time as the current direct route and in order to do that, I had to go via Lincoln, at least no reversing at Sheffield. The current route is the slowest possible route. If the route through Bakewell ever reopens that will be physically the shortest but unlikely the fastest.

Another possible route would be Liverpool->Chester->Crewe using the Halton curve. that would add about 35 minutes to the Liverpeel Crewe section.

The British Rail Class 755 would seem ideally suited to the journey, the current trains are British_Rail_Class_158.

CrossCountry would be the ideal operator for this.