The places nearby are a Premier Inn IIRC. Its not too far though from Romsey and Southampton so plenty of places nearby to stay. Also if you are planning to stay for a few days maybe try going into Portsmouth as you have the Dockyards/ Spinniker Tower etc to visit the next day. I don't think there is any accomadation on the site inself.
Peppa Pig world and Winter Wonderland are fab though.
Try http://www.thenewforest.co.uk/accommodation/ There are loads of places to stay near Paultons Park and Peppa Pig without having to go as far as Winchester. We stayed at an excellent B&B in Cadnam, with a fab restaurant immediately opposite. We didnt even have to drive to Peppa Pig, because the B&B owner ran us up there in her car! Try Twin Oaks Guest House they couldnt do enough for us. Happy hunting.