This is caused by the Papillomavirus, a DNA virus that is more normally associated with parrots and macaws. For those with an abnormal fascination with the virus, more details can be found in this paper.

This Chaffinch is one of the most extreme cases we've ever seen, and was caught on 15 March near Ripon. The extent of the growths is incredible, and the bird obviously has difficulty walking and presumably feeding normally, as it has very abnormal claw growth on its feet. The virus has been noted in the Chaffinch population at the site for several years, though the majority of birds coming to the feeders in the winter still have 'clean' legs. One adult male, though, that had regularly been caught at the site had lost one foot due to the virus.

Thanks (I think) to Yvonne Elvin for the photos and to Jill Warwick for passing them on.