For some reason, these images simply don't upload, and after some playing, I got some help from Firefox:
This had me off looking at the HTML for the page, which revealed the problem. Some time ago, LinkedIn implemented HTTPS on all its pages, advising users, quite correctly, to switch over to this as soon as possible. This means that the URL for the "edit company overview" page will look like this:
However, within that page, the form to upload images has the following action URL:
It is this mismatch between secure and not-secure that gives rise to the message from Firefox, and it turns out that this interferes with the image upload process.
Therefore the workaround for this is to switch off the option to use HTTPS when possible (Settings -> Account -> Manage Security Settings). The dialogue looks like this:
Untick the box, save, and then navigate back to your company page, making sure you are now using HTTP and not HTTPS, and you should find that the images now upload correctly. When you have completed your work, you should of course return to the security settings and switch HTTPS back on.