The nearest Household Waste Recycling Centre to Chilworth was, until 8th December 2016, the one in Woodside Avenue, Eastleigh. Its replacement is much more convenient for Chilworth residents, being behind the BMW dealership in Chandlers Ford. Location details, and opening times, can be found here. NB It is now necessary to make an appointment before driving to the site, and you must also pre-register the number plate of the vehicle that you intend using.
It used to be a toss-up between the Eastleigh facility in Woodside Avenue and the one in Endle Street, Southampton (near the Itchen Bridge) but the latter closed in December 2011. The Southampton replacement is at First Avenue, Millbrook (hard by Dock Gate 20) and information about it may be found by following this link. Note that this also comes under Hampshire CC these days, and so the same booking and registering regime applies.
Heading northwards, it’s not too far to the excellent facility at Bar End, Winchester, just off M3 J10. The beauty of using this one is that the Emmaus Hampshire live/work community is right next door. The formerly homeless residents of the community make their living from selling goods donated to their showroom by the good folk of Hampshire. There’s always a stock of good quality furniture and other items for sale, and a coffee shop besides. Well worth a visit!