<Australian passport requires a Schengen visa for the member countries >
Not true - they can visit as a tourist without a visa, as they can virtually every European country with very few exceptions.
Getting back into the UK <could> be a problem with an Australian passport if OP is resident here and it contains no evidence to the effect they are entitled to be so. (That is, they relied on their British passport, which they don't have, as evidence of their right to live in the UK). They <could> be refused readmission as a tourist (because they are not), and refused readmission as a resident (as they have no evidence of entitlement)