The 3-post thread on this topic in "Infrastructure & Stations" starts with a reference to 20 minutes between Liverpool and Manchester. Given three intermediate stops and the poor alignment of the old LNW Low Level line at Widnes and especially through Warrington is this even a remote possibility? Wouldn't the expensive new route actually be no quicker than non-stop via Chat Moss (especially if the latter saw some improvement)?
The fundamental problem of the Chat Moss is that a train running non stop will just run into the back of the stopper in front, because of the density of stations.
I think you could probably do Liverpool to Manchester with at least two intermediate stops in 20 minutes, but it would require a high performance trainset on a new alignment with a top speed of 260km/h or better. ie. a Shinkansen.
With three stops, even short periods of non stop running will likely make the objective functionally impossible.
EDIT:
Even drawing straight lines through Liverpool, somewhere near Liverpool South Parkway (where actually is Gateway?), Warrington and Manchester Airport gets a distance of 63km. That would require an average speed of nearly 190km/h to do it in 20 minutes. That's pushing it, even with one stop.
If you abandon the idea of serving the airport it is much more tenable.