I have made two discoveries since posting my question. One is that (at least in my case — iPhone 3G, T-Mobile [now EE] network) it is the duration in seconds, not the number of rings, that is predetermined. Thus if a short ring-tone is selected on the phone, it will ring more times than if a long, more musical one is chosen. That doesn't help much.
For the other, I went to a local EE shop. They innocently explained to me that, especially if one is on Pay as You Go (as I was), the network company deliberately restricts the ring time because if a call goes to Voice Mail and one then listens to it, one is charged for doing that — and charged even more if one replies — and one's own network cashes the charge, whereas if one picks up the call, it is the caller's network that gets the money. They persuaded me to change from Pay as You Go to a monthly package. That has slightly increased the length of time the phone rings — it is now 13 seconds — but only slightly, and there seems to be nothing further that can done.