Typically, despite fiddling about with the config for ages before posting this question, the moment I posted it I found out more details. It's not down to the version of IE on the client, I am actually getting the problem on my client running IE8 as well. It's something to do with the VSM system set-up that is causing it. Quick Launch on our Dev system works perfectly on all version of IE, it's just our Live system that it's not working on.
Given that Live is configured exactly the same way as Dev, then I guess it simply requires all of the services to be restarted and/or have an IISReset on the server. I'll try doing this out of hours this evening to see if it resolves the problem.