Thanks to you all for all the advice and suggestions. Here's an update on not much progress so far...
Unfortunately, the new batteries failed to solve the problem and, having swopped the units to check if there was anything in the socket causing the problem with the one unit, (my logic...if, indeed, that could be called a logical action!, the alarm started shrieking constantly, on battery only. I have now assumed that there is a fault in the unit itself! Why this should randomly happen after seven years, out of the blue, with nothing apparent to trigger it, who knows? Perhaps someone here does.
I am hoping that the company that supplied them will still have the same alarm, or one that is compatible with the base plate, so that I don't have to start drilling new holes in my fragile ceiling - probably a faint hope after seven years.