FIXED IT!
In March I'd installed a PDF reader/printer that interfaced with Firefox - Foxit Reader. It got in the way of the clipboard on launch of anything Office.
I found a very old thread on an Office forum that talked about this problem and "add-ons". It suggested loading the clipboard, then starting Word by typing winword /a on the start button, and testing to see if it would paste. It would.
Foxit was the only add-on in MS Office (the 2003 version, by the way, still the best in my view). Removed that, all is well.