Thank you @Peter_W for replying.
Both handsets are paired and have been working well for over 12 months.
After I posted the original query I continued searching online and ended up back on the BT community website (which is where I had begun my search). I eventually found a couple of posts which referred to the problem and suggested the batteries in the handsets were not holding their charge very well. Apparently, when the speaker button is pressed, the demand for additional current cannot be supplied and causes the call to drop and the handset to attempt to synchronise with the base. The original batteries supplied with the handsets looked cheap and rather old and so this seemed a reasonable explanation. Mr Amazon kindly supplied the 800 mAh Li-Mi AAA batteries today and I inserted them in the handsets.
So far, so good. I've used both handsets to make a couple of calls and then switched to speaker once connected. Everything worked as it should and so this appears to have fixed the problem. I'm just surprised the problem manifested itself on both handsets at the same time.
Thank you once again for taking the time to give me your advice. It is appreciated.
