Interesting. The 1N4007 has a 1KV reverse breakdown voltage but the 1N5392 is only 100V. A 1N5399 would have the same reverse breakdown voltage as the 1N4007. The 1N539x diode does have a higher voltage drop though at 1.4V for 1.5A. Both are standard recovery types. RL255G is rated for 600V with a 1.1V drop and 2.5A capacity. I would ask Mesa if the reverse breakdown voltage of the 1N5392 is sufficient.
Since the 1N5392 can support up to 1.5A, the 12V supply will not be choked off as much due to the limitations of the 1A diode. Not much else will make a difference. The voltage regulator also has limits on its ability to maintain regulation at a specified current rating. Once the parts get taxed with too much load, they tend to fault or fail. Most of this is due to thermal degradation.
From what I saw on the schematic, V1, V2 and V5 have the heaters supplied by the 12V power supply.
The rest are supplied by 6.2VAC: V3,V4,V6, V7 and all 4 power tubes. It is the green wires from the PT. the Green/Yellow is a center tap on the winding connected to chassis ground. It would not be difficult to see if the heaters are active on V7, you will need to remove the tube shield to see if the cathodes or the heater element is glowing. Also, if there was no heater voltage to that tube, it will also remain cold to the touch. It that is not the issue, and you have no signal at V7 control grids, the issue could be part of the strobe mute circuit. The only schematic I have seen is missing the switching circuits and strobe mute circuit that controls M1 and M4.