british parents, children born overseas

Both my wife and I are british and have now moved back to the UK.
Our children born in Australia have Australian birth certs. and passports.
When we arrived in the UK they got a stamp to stay in the UK for some months as tourists.
Must they get British passports or can they stay in the UK as long as they want and fly on just the Australian passports?

Both my wife and I are british

Are both or either of you British born?

They should have a visa to live in the UK or British citizenship, depending on what type of British citizenship either of the parents have.

We are both british, wife was born here,
Is it illigal for them to overstay the visa, if they are eligible for a British passport and must we actually get a British passport?

Australia allows dual citizenship hence my confusion as to why you might not wish to apply for British passports.

A British passport would make the most sense for them. There are restrictions on visitors and access to certain things, NHS is one that comes to mind as visitors are not entitled to free NHS and GP surgeries now have to take proof of eligibility and future work as they don't have the proof of right to work or study with home fees.

If you don't wish for them to have a British passport and they appear to be British by Descent, you could apply for a Certificate of Entitlement to be endorsed on their Australian Passports.