Barnard Castle (the castle, that is, not the town of the same name, famous for its opticians) is expected to re-open in early July, along with Middleham Castle and Mount Grace Priory Gardens. Admission is expected to be by booking only; look for updates on the English Heritage website.
English Heritage's Richmond Castle looks as if it may open mid-August.
Welcome to Richmond Information Centre's website, the best source of information on this wonderful town. Take a look round!