Question Author
Just done that, Gness.
They were really helpful. Apparently, they are aware of the problem. It's trichomonosis, a parasite which affects greenfinches, pigeons, doves and some birds of prey.
They have recommended that we stop feeding the birds and keep the bird baths empty. The disease is passed on through contact. We need to try to disperse our flock of pigeons. There are likely to be more poorly birds in the coming weeks.
Oh!
Thanks for the advice, Gness.