Hey guys!
I had problem with reverb on my Mark III head too, it did not work at all.
So I ran through all the hints mentioned here - changed the reverb tube, checked all wires and grounding, everything looked fine, but still no reverb.
Then I checked cinch connectors and finally found the "needle in the heystack" - problem was in cinches on white wire, which connects chasis out and reverb tank input - these are the ones with wings (no shielding).
The weak point at this old cinches are the tips - white wire is put inside and soldered at the tip to make a connection between wire and rest of the metal stick. In my case the wire wasn't soldered well to the tip, or it got loosen, so there was not connection with the cinch stick - even I have checked the wire's impedance with digital multi-meter before, I did not found this tricky thing, because wire touched the tip of the cinch accidently, so it looked fine.
I have re-soldered the tip with huge amount of soldering tin and let it drain into the tip and now my reverb works again.
To check if this is your possible problem just unplug the cinches from chasis and use any multimeter for impedance between the tips of both wires - there should be impedance about 200 ohm. If your meter does not show this value, then for 100% one of the wires is broken or you have a same problem as I had.
I hope this hint will help you - if you are not sure, do not hesitate to write me post to my PM, I can make a picture with detail of the tip which cause the problem and send it to you.