I think that Chester was the last zoo to breed the great indian hornbill. Hornbills in general are notoriosly difficult to breed . Some species, like the rhinoceros and south Sulawesi have only been bred once, while others, like the Von der decken's hornbill are bred regularly.
I know nothing about Paulton's park sorry.
I doubt that many hornbills are held in private collections. If any are kept, I doubt that they'd be any of the species that you mentioned, as they are all quite unusual in captivity. Only a handful of zoos have been sucessful with the writhed, so I doubt that any private collections would house them. The rhinoceros hornbills are only kept at Paignton and Chester in the U.K, and the Great Indians are slightly more common.
If any hornbills are kept in private collections, then they'd probably be a species like the Yello billed hornbill, or the ground hornbill, the more common species in captivity.
Have yellow or Black casqued hornbills ever been kept in the U.K? They are both so unusual.