OK. I seem to have got it working. Here's what I did - tell me if it's a bad idea although I should point out that this is an intranet so I'm a little less concerned about security...
Go to Control Panel->Administrative Tools_>Services
Select the Apache service and hit Properties
On the Log On tab, click 'This account:' instead of 'Local System account', and then find the User account of Windows user who's normally logged on to that terminal
Restart Apache
I also amended the user info in the httpd.conf file, but I'm not actually certain that that was necessary.
If there's a better solution, that can be explained in words of two syllables or less, I'm all ears!