Here are your parts
Replacement parts for WHIRLPOOL REX5634KQ1 | AppliancePartsPros.com

Here is the wiring diagram
http://www.servicematters.com/d...0Sheet%20-%208528187.pdf

Have you checked the voltage?
Try flipping the breaker off/on slowly a couple times.
Better yet measure it with a meter.
Sometimes you can loose half the line without actually tripping the breaker.
Your timer and heater both require the full 240 volts.

Your heating element should not read 0 ohms, should be 10 ohms approximately.
Looks like you may be using too high of a meter scale.
Also be sure to disconnect one side of any device you are measuring. This prevents you from reading an alternate/parallel circuit path.

Try running the unit in timed dry.
Does the timer now advance?
If yes then you know that the unit is getting 240 volts, then odds are high that you have a heater problem.

The way it works in Auto cycles is that when the thermostat shuts off the heater, the timer gets voltage through the heating element.
When the heater is on, it drops (uses) all the voltage so the timer does not advance.
In timed dry the timer is wired directly across the 240 volts and is independent of the heater.