Thanks Barry, I had come across that issue that other people have been having, and after you suggested it I tried playing with those settings, but that didn't fix the issue. Obviously I couldn't try every combination of numbers, but I tried adjusting Ray Threshold, Ray Depth and Reflection Depth individually between 0 and 100, and the main difference was windows either being black or clear.
I was also able to get the closer windows to be clear, and the far windows to be black, which suggests that when they show the blue behind them, it should actually be the background image, and that it is another problem that is causing the background image not to show. And the blue is actually the 'Sky Visibility' from the Environment Settings - as in, if I turned Sky Visibility on under 'Compositing', that would become the background for the whole render, and the windows would look the same as they do in the image above.
Actually, I've just narrowed it down a bit, because when I turn off Sky under Environment > Physical Sky in the render settings, the background image comes through as it should. That is what I did originally when I said the render worked correctly with the stock scenes, because I thought that was how you used a custom background image. Then I remembered the setting I needed was under Compositing.
So based on that, does anyone know what I could change to get the desired result? Obviously turning off Sky completely changes the lighting, so I don't want to do that.