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So I am wondering if the cause of this mystery is an (intermittent) electrical fault in the clock motor, which caused it to depart (occasionally) from synchronous mode. I am aware that shorting in a multi-pole synchronous motor can cause over-speed. However, the timeswitch - from a well-known UK supplier of central heating controls for the domestic mass market - comprises a "2-pole motor and integrated geared output assembly. [The design] was devised by clock-makers in the Black Forest in the late 1990’s, and is now copied by many producers worldwide (including our Chinese supplier)."
The installation is conventional (to switch a domestic immersion heater). I did not watch the clock to see if the time gains were incremental or spasmodic - but then neither do I watch paint dry!
As I am not familiar with the design of miniature synchronous motors for clocks, the observed behaviour puzzles me - as it does the supplier. All and any thoughts on the matter will be welcomed.
So, was their time clock really a clockwork clock with an auto-rewind?
Certainly in the 1970's we had a battery (D-cell / U2) powered kitchen clock which was clockwork powered with the spring being rewound at intervals by a small motor powered by the battery.
But the fact it kept good time on test at the supplier is very telling. You may yet have something a bit odd with the harmonic content of your mains supply , but only marginally and or occasionally, or the replacement would do the same.
PS I recall an electric clock without a crystal and dividers, but an electric escapement with a PNP transistor and a driving and pick up coils cunningly placed around a pivoting magnetic, so the transistor pulsed at just the right tiem to keep it going. Until it stopped doing so, which is how I saw it in bits. How cheap IC fabrication change things, all 32KHz crystals these days, dividing to one pulse per second.
As mapj1 suggests, the clock mechanism looks simple and straightforward. Photos of original when examined by the supplier: