Thanks for your help, Old Pendant.
Another thing that occurred to me was that the Met Office script actually contained a document.write() - this is not Ajax friendly so it wouldn't re-render the widget.
I downloaded the script and saved it locally, changing the document.write() for html() and hey presto!, it works just as I want it and doesn't use iframes.
I don't know why, but I have something in my mind that makes me think iframes aren't great (accessibility?, not sure), so I am happy to have found a non-iframe solution!
Working version at http://www.admin.webstu.co.uk/weather-test.html