Hello, we have now resolved the issue, so this thread can be closed, but our solution can perhaps be useful for other users experiencing something similar in future.
The reason for the widget failing to load data was that we also had a small counter script running on our homepage. It appears that the counter provider has now permanently disabled the service. The failure for the counter to load also resulted in the failure of the Twitter widget to load data. I’m not sure why there should be any connection between the two, but there was. As soon as I removed the counter script from the webpage, the Twitter widget started working again. And this although the Twitter widget is located on lines 163, 164 of the source code of hour homepage, whereas the dysfunctional counter script was only located towards the very bottom of the webpage, on line 243 (of 250 lines total) of source code of our homepage. (And no, the two scripts were not part of the same HTML table – they were located in distinctly and completely separate sections of the homepage.) In any case, this mysterious issue is resolved now.