You most certainly can fly on your UK PPL. You will have to go into the CASA (Civil Aviation Safety Authority) offices at Bankstown airport (That's the big GA airport just west of Sydney). Answer a few questions and show them your PPL and logbook, pay your $50, and you get a lifetime Australian endorsement that's valid as long as your PPL and Medical are valid.
There's lot's of different clubs to fly with at Bankstown, I flew briefly with Basair, and found them to be very friendly and professional. Australian GA is incredibly well organised (Could teach the UK at thing or two) so you shouldn't have too much trouble picking things up. Have fun!