How do I get from Glasgow to Alton Towers the cheapest way?

I have called Traveline Scotland and they've advised me to get a coach from Buchanan bus station to Stoke-on-trent, then another to Alton Towers, but does anyone know if this is this the cheapest or even only to travel? I think the train goes straight to London and I don't fancy a trip from there all the way back up to near Derby!