Seems to be working again now. I don’t know why.
I deactivated the plugin on my live site. Then I went to my development environment and upgraded to WordPress 3.5.1 (I know, I should have done this first before upgrading the live environment!). Having done this, Twitter Wings was still working. So the problem probably isn’t caused by the upgrade to WordPress 3.5.1.
So I went back to live environment and re-activated Twitter Wings and it worked. As I say, I’ve no idea why.