Is this an old 50's 5E4, or a modern copy?
I see possible headaches in your future, as you sort out the wiring to function like you think it should.
Let's say it's an old 5E4. The filament wiring was done by running a single wire to each tue socket and the pilot light, with the other side of all (including the PT filament winding) connected to the chassis in some way. Reworking the filament wiring will likely prove a frustrating experience, because you will almost certainly miss the tricky grounding of one side of the pilot light. I speak from experience with these particular amps.
If it's a modern copy, you may have some difficulty sorting out how the modern builder attempted to copy Fender's 50's-era techniques.
If you get no voltage reading across the 1 ohm resistor, then it is probably being shunted by some alternate path for current to flow to ground, rather than going through the resistor.
The problem here is (assuming everything is wired perfectly, and according to the old method) that because the pin 2 side of the socket was wired with a short pigtail to the chassis, you might have difficulty breaking the connection over to pin 1 (not needed, except maybe in a metal 6L6) and pin 8 (the cathode). You could run the resistor from pin 8 to pin 2, which is then connected to the chassis, but the goal of that is to avoid soldering to the chassis (if that is how the connection is presently made).
A better choice would be to run the 1 ohm resistor from pin 8 directly to ground. My thinking with that is to not have the cathode connected to the filament circuit if possible (not really a problem *if* wired 100% like 50's Fender, but not ideal). If you attempt to solder to the chassis though, it's very easy to create a high-resistance cold solder joint, and still look sorts like a good solder joint. If you have a cold solder joint to the chassis and push against it with a flat screwdriver, it will pop right off with only mild pressure (I found that out the hard way by making poor solder joints to the chassis in a couple amps). If this is the case, and you still have the connection from pin 8 to pins 1, 2 and then on to ground/chassis, then the good ground through the filament ground at pin 2 will shunt current away from your resistor, leaving no reading.
So if you have no reading (in mV's) across the 1 ohm resistor, and the tube is actually passing current, then there must be some wiring error or shunt path for tube current to flow. The trick is to figure out where the problem lies.