I've sawn off the obviously dead branches reducing the 20ft tree by half. I saw no signs of eggs but parts of the bark were canker-like and flakey, and there was dark staining on the sawn ends. I think this will be her last year sadly, after seventeen years.
PS Since my inital post I've seen three Paulownia tomentosa trees in Battersea Park and an even grander one in Eccleston Square. It's not called the Foxglove Tree for nothing. Fabulous!