I've copied, deleted and then reloaded all the image files. All now working in Firefox, Chrome and Opera. I don't have Safari installed but if anyone does I'd appreciate a shout as to whether that's ok too.
IE 11 (maybe in earlier versions too, feedback welcomed) does not work for anything except png files, which is why a minority of avatars are showing up, but many aren't.
I'm not suggesting anyone change their browsers, that's a personal choice and I'll be persevering with IE myself, but at least now I know it's not a problem with site integrity or the image files themselves.
Very recent notes on the SMF support forum, which I'm monitoring daily, are suggesting this is may be a bug within the forum security checking code. There's no work-round offered yet which doesn't compromise site security to a greater or lesser degree, so I'm loath to apply the hacks suggested so far.
I'll update when more news available.