I received the new oscope. It works!
Here's some of the waveforms. Someone on another forum recommended doing some tests with TR814 and TR815 disconnected so these are with them removed and all wiring on primary and secondary attached as normal.
This before connecting, an unloaded signal. The signal, by the way, is off a 555 wired up as a 20kHz PWM, with roughly 3.5v peak. I don't know how much power it provides but it's coming off pin 7 so don't think it's that much really.
On wires 13, 17 and 16.
On wires 12, 14 and 15 (14 is marked as 11 in the schematic).
I don't really understand what the four primaries (or are they centre taps) are doing in the circuit but the top half is totally loading down my signal to nothing while the bottom half is not affecting it so much.
Now with the scope powered up....
This is the oscillation on Position 13. It's a lot slower than I expected. The amplitude is so small I have to use AC coupling to see it. The oscillation is 100Hz so is probably power supply ripple or something.
And position 14
It's stable like this and only wobbles a very small amount. Nothing is heating up.
Swapped in a new (probably old stock) MJE520 at TR813 (to replace the doofah BD435) and there's no change to oscillation speed or amplitude.
Looks like power rail ripple, it's at 100Hz and 0.4v. I wonder if the main reservoir cap is doing a good job.
I will connect up TR814 and TR815 tomorrow and take a look at the waveform then.