Good idea.
A very, very cursory glance at the timetable suggests that might actually work fairly well.
On the Up, you'd be chased out of Warrington by the Edinburgh-Birmingham-Euston about 8 minutes behind, so you'd probably have to go Slow Line at Acton Bridge to be overtaken by it (meaning the one of the hourly direct Warrington-Crewe services would overtake the other)
On the Down, you'd have to depart Crewe somewhere ahead of the xx33 'fast' Birmingham-Liverpool, and behind the xx10 Euston-Birmingham-Scotland.
You'd get about a 10-15 minute turnround at Crewe, using only one extra diagram.
Only trouble I can see is you'd need to be arriving at Crewe into bay Platforms 9 or 10 about the same time as the xx10 Scotland is departing Platform 11 - can't recall if these moves can be done in parallel or not.