Theory:
The wobble appears to be from an unsteady power supply feeding the magnetic deflection plates.

I'd suspect bad/out of tolerance high voltage electrolytic capacitors, and if its not that, I'd check voltage regulators and the high voltage board's transformers.

Rationale:
A CRT monitor works by guiding a cathode ray (a beam of electrons) created from a hot cathode at the back of the monitor's yoke using a high voltage transformer (the flyback), and a high positive voltage collection system near the front of the monitor's tube. (the line that drives this positive front voltage looks like a great big suction cup stuck to the top of the tube, with a big fat wire on it.) This ray is deflected by very precisely controlled electromagnetic coils on the back of the yoke. They look like a big copper wire collar around it. Electrons are kicked off the hot cathode through thermal emission, and are accelerated toward the collection plate through electrical attraction. Since the electrons have a charge, and are moving, they exert a magnetic moment via the right-hand-rule, which the deflection plates exploit, to bend the path of the beam with very precise degrees and intervals of energization.

When these magnets are not so precisely controlled, the beam's path is not reliable, and you get these kinds of funny artifacts.

There is a short list of things that can cause these to not be precisely controlled:

The high voltage electrolytic capacitors have outgassed, and dont have the proper capacitances any more, making any resonant circuits made with them not operate at the correct frequencies anymore.

The voltage regulators controlling the source voltage for the coil control circuitry are faulty, or out of tolerance, causing odd behaviors.

There is EM noise from a failing coil somewhere in the high voltage board, and the yoke coils are picking this up like an antenna.

There is a cracked solder join for the deflection yoke, and the conductivity to the yoke is not reliable.

etc.

You will need to get in there with some high voltage capable oscilloscope goodness to verify that all the power signatures are reliable, and you'll need to inspect all the caps very closely.

We had similar issues on new Samsung CRT monitors in 2003 due to the office room being next to a strong electromagnetic field. We moved to a different office and the wobble went away.

That looks like 50/60 Hz mains hum, but it's pretty bad. Either the PSU in this thing still has some serious issues, or it could be an external field causing that. Any cables going right next to that monitor? CRTs are mostly affected by things closer to the screen, the neck area and back should not be sensitive at all, if in doubt try moving the monitor somewhere else to test, like the middle of an otherwise empty table. Make sure both the monitor and the signal source are powered from the same socket, or there might be issues with that too.

Deunan wrote on Today, 08:41:

That looks like 50/60 Hz mains hum, but it's pretty bad. Either the PSU in this thing still has some serious issues, or it could be an external field causing that. Any cables going right next to that monitor? CRTs are mostly affected by things closer to the screen, the neck area and back should not be sensitive at all, if in doubt try moving the monitor somewhere else to test, like the middle of an otherwise empty table. Make sure both the monitor and the signal source are powered from the same socket, or there might be issues with that too.

60Hz mains hum can easily be identified by setting the refresh rate to 60Hz. The wobble will be very slow in that case. 50 Hz mains hum may appear to "stand still in certain places" at a refresh rate of 100Hz. If you get a nearly stable or very slowly moving image at 100Hz or 120Hz refresh rate, you have hum at twice the mains frequency. Interference at mains frequency (50/60Hz) is likely from badly shielded cables (or the electric train supply wire behind the wall), whereas interference at twice the mains frequency is a sign of bad capacitors after a mains rectifier. If the supply is a standard switch-mode supply, that's the 400V electrolytic near the AC input, and if that supply has PFC, 100/120Hz mains hum can be caused by a faulty PFC circuit.