The auto transmission won't engage any gears in my 2009 Ford Falcon FG

This transmission has a finite lifespan and when they die, they sometimes just stop working as yours appears to have done. An automatic specialist will know the telltale signs of this and should be able to diagnose the problem.

But don’t give up hope; you might find the problem is simply a low fluid level in the transmission. Low fluid can certainly provide the symptoms you’re seeing. However, if that’s the case, then you’re chasing a leak, because these transmissions are a sealed system and shouldn’t need periodic topping up.