Taking the train is a fairly doable option (having done it myself). Take an early morning train from London Euston to Stoke-on-Trent, then catch the shuttle bus from there to Alton Towers. Do the reverse on the way back, I recommend booking a few weeks in advance as the train tickets are significantly cheaper, also travel on a weekend as weekday morning trains are very expensive.
When I did it I was staying for a few days but I can't see why it couldn't be done for just a day (except being a bit tiring!) Hope this helps.