The problem I see is that your 24vac transformer isn't really doing anything in your system. It might be there for an accessory that you don't have, but it's not hooked up. If it was doing anything, there would be a wire connected to the R and C terminals, and that is not the case. Use a multimeter to measure between the R and C terminals to see if it is outputting 24v - if not the transformer might not be getting power at all.
In a typical HVAC system, the thermostat acts as a series of switches that connects the R wire to Y, W, and G to activate 24v relays in the system that turn on the compressor, heat, and fan respectively. Assuming the thin white and red wires in your pictures go to the thermostat, that's not the case for you. Your heater just needs the white and red wires to be connected, but they don't carry voltage. It operates like a "millivolt" system and expects the (mechanical) thermostat to only provide a contact closure to connect red and white when heat is needed.
This is not strictly compatible with the Nest thermostat, but it can be made to work with the help of a relay. In the diagram, the connections on the nest would be: Red - R, blue - C, and black - W (meant to be the white wire, but I used black).
You have all the items on that diagram except for the relay. You will need to find a "normally open" relay with a 24vac coil. It is a pretty common item if you know what you're looking for and here is an example. (I just noticed, one of the reviews on that relay also talk about using the nest with an old millivolt heater.)
The diagram for that relay is:
Where terminals 1 and 3 are for the coil, and 2 and 4 are normally opened (note the diagram is flipped compared to the one above).