[RJMiranda]

By all means use an isolating xfrmr. Connect the radio set to it (the radio is the one that may have voltage at the chassis, the scope is safe). The thyratron circuit is not fed from an isolated supply, for its plate supply comes directly from the AC line, and the tap on the secondary of the filament transformer is connected to the line also.
I would connect the scope ground to the control receiver´s chassis. As you will be measuring AC and not DC, this point is equivalent to the hot "off-center" tap of the filament transformer. It is safer to ground the scope at the chassis and not the center tap.
With one channel at the 2A4G´s plate and the other at the grid, the positive-going half-cycle at the plate must coincide in time with the negative-going half-cycle at the grid.
Some time-displacement will exist between waveforms, because of the reactances of the grid capacitors and coil, and at the plate the relay coils, RC series network in parallel with the relay, and RF choke. But I think that, generally speaking, the voltage at the plate will be positive-going while the grid is negative-going.
If this is not so, either a) reverse the primary leads of the filament transformer or b) swap the conductor from the second relay coil to one side of the AC line with the conductor from the tap on the secondary to the other side of the AC line. That way the phase will be shifted 180º.
The battery stopped the thyratron firing because it put an additional negative voltage on the grid, even if the AC phase was wrong. But if the AC phase is OK, then it may be that the thyratron performance has degraded, and so it needs extra negative bias.
The receiver-amplifier works OK, it seems, because with the battery inserted you could control the relay from the remote box.