Finally got my TCP/APN settings working. I fooled my third-party application into opening a test page which I configured to capture the IP address. Here's what I've found:
* Apps which don't require APN use a RIM IP address, suggesting that the data goes via BlackBerry.
* Apps which require APN use an O2 (my provider) IP address.
I will telephone O2 and find out whether this is included. Though, judging from other comments in this forum, they probably wont know what I'm talking about.