After replacing my 2001 Holden Commodore's transmission, the engine no longer turns over.

You need to go back to basics here. Check that the battery is connected firmly and that all the connections and low-voltage leads that power and trigger the starter motor are in their correct places and receiving power. Don’t forget to check for earth straps and cables that might have been left off in the reassembly process, too.

If the car is getting power (and the dashboard lights suggest it is) then there’s a problem between the ignition key and the starter motor. Tackle it methodically, checking for power (a test lead or multimeter will be invaluable for this) as you work your way towards the starter motor and you’ll probably find it’s something really simple and basic. Make sure the fuse or relay for the starter motor hasn’t been fried during the gearbox swap, too.