Pick up the bird, sheet and all, and place in the box. Gently try to remove the sheet, leaving the bird in the box. Any bird found with a cat should be examined by a Wildlife Rehabilitator immediately. All wild birds are protected by law. Thank you for your assistance in preventing avoidable injuries to wildlife.
Birds have a very poor sense of smell.
This means young birds can be returned to the nest. The parents will continue to care for them, but only if you keep your distance. If an entire nest has blown out of a tree, the nest and babies can be put back up in the tree. Baby birds are very difficult to hand-raise. Their nutrition, temperature and housing requirements are very specific.