You can fly PPL (SE or ME) on your Australian PPL in the UK so long as you're happy to stay day/VFR in UK airspace. Going IR / night / overseas will require a JAA (becoming EASA) licence.
Billins Air Services at Cranfield are worth a look.
Obviously Turweston, although I don't personally know the schools there.
You could try north a bit and look at Brooklands Flying Club at Sywell, which is just the other side of Northampton from the M1 Junction.
G