I've now been in the UK for just short of two months and finally managed to see a GP today to get a referral to see an oncologist to be sent off to have my next scan (don't hold your breath). I am exaggerating a little about the delay, however, since registering with my local doctor required the completion of my university matriculation (enrolment). Although people are quite willing to complain about the NHS, I haven't yet seen anything to make me think public health ought not to be publicly funded.
One difference between GP care in Australia and the UK is that here you have to register with a single practice in your area and can only receive NHS funding to see a GP at that practice (until you change your registration). This means that patients can't shop around for the right diagnosis (or gather multiple prescriptions). It also means that all your medical records are collated in one place.