[WaveyDipole]

Further investigations show a number of resistors gone high in the Miller Transitron timebase circuit but the key discovery was that the installed VELocity control pot is not only faulty but and incorrect part has been fitted at some point.

Once slackened off the chassis, a marking of 5000Ω was revealed on the pot body but the circuit diagram indicates that it should be a 2MΩ part. Perhaps someone mistakenly thought is was a 5000kΩ (5MΩ) pot?
Clearly a 5kΩ part would not be a suitable replacement for the 2MΩ pot. Its track resistance, with one end disconnected from circuit, actually measured just over 100kΩ and 2-3 megaohms between wiper and track except at either end where contact is made with the end stop where it would read 100kΩ again. It seems that the track must be fried somewhere along its length and the odd resistances are likely the result of soot residue.

Whatever the case, the pot needs to be replaced. I do have a 2M2 pot to hand but the thread diameter is rather wider than the aperture which I prefer to to widen to that extent so will have to source something a little smaller. In the meantime though, I hooked it up temporarily to test and its a totally different experience. Whereas previously waveforms were an un-synchronised free running blur most of the time, stabilising only very occasionally with a lot of fiddling, they are now stable and can be viewed, contracted and expanded with ease. Occasional artefacts do still occur while adjustment is being made, but as indicated by David and Adrian's comments these can now be accepted as normal.