There are regular connections from Manchester Airport to Manchester Piccadilly Station (6 trains an hour, so every 10 minutes)
There is a train going from Manchester Piccadilly to Stoke On Trent roughly every 15 minutes, so that isn't going to be a problem
The times of the bus from Stoke to Alton are 0710, 0820, 0915, 1010, 1110 1545, 1730, 1830, 1930, tickets for bus travel cost approx £4/5 or I believe you can buy your park tickets on board the bus and get travel for free, depends if you have any 2 for one deals, don't know if they accept them on the bus.
You will have to get a train from Stoke on Trent to London Euston Station then get the Underground over to St. Pancras for Eurostar, which is one stop on the Victoria Line, shouldn't be too bad (ie, not very easy to get lost!)
P.S. Good website for planning train journeys etc..... www.thetrainline.com
although they are not quite ahead as september yet