Thanks for the suggestion, old pendant.
I don't know why, but I have it in the back of my mind that iframes are the best solution (accessibility?), so I happened to download the script and have a look at it. Turns out that it contains a 'document.write()' - not AJAX friendly! So, it's no wonder that the widget wouldn't re-load!
Anyway, I have downloaded the script and changed the document.write() to an html(), and refer to the script locally instead of externally.
All now is fine and works just as I want it to!
Working example at http://www.admin.webstu.co.uk/weather-test.html
Very best wishes,