You said that the timer turns on so I presume you mean the display is on. Can you alter any settings on the control panel using the buttons, you didn't say?
The light is working (you didn't mention the fan), the control panel display turns on and you have tested the door interlock switches, I suspect that there is a problem on the control board - GE part #WB27X11080 (supplier example only).
Here's an image of the wiring diagram for a GE JVM series microwave. It is not for your exact model but the wiring diagram and the image of the control board part I linked (which is for your model) seem very similar. By that I mean it has relays that have similar designations regarding their function etc.
To get the light, fans and the turntable to rotate and also the HV circuit ( magnetron circuit) to work relays on the control board have to be operated to connect power to them. These are operated depending on what signals the control panel has sent to the control board.
It could also a faulty contact in the start button on the control panel GE part #WB56X20404 not sending the signal to the control board which would operate the main relay to start the heating process.
IF there is a 16 wire ribbon cable between the control panel and the control board you can use an Ohmmeter (high resistance scale) to check for continuity between the pins as shown in the table in the image to see if the Start button is OK when operated. As I mentioned earlier this is not for your exact model but hopefully it will be similar and relevant.
if the control panel tests OK (check some more buttons) then its the control board.
Be safety aware when working in microwave ovens. Disconnect the power to the oven and then correctly discharge the HV capacitor as soon as it can be accessed, before performing any other work inside the unit
(click on image)
Are you testing the interlocks with the door open and the wires still connected to the switches ?
If so you may get those resistance readings as you're measuring other components not the switches.
With the power disconnected and with the door shut and the wires disconnected from the switches, one should test short circuit i.e. 0.00 Ohms (primary interlock) and the other one should test OL or open circuit (monitor switch), when measured directly across the switch. With the door open it should be reversed. The primary interlock switch should now test open circuit and the monitor switch should test short circuit.
Here's the JVM series troubleshooting flowchart that may help.
(click on image)