No one else has answered this yet, so I am going to answer based on what I was told at the visa processing centre in London. I'm aware that you may no longer be requiring this assistance, but perhaps it can be useful in the future for somebody else.
The answer is maybe (see below), but you should try to speak to TLSContact (the contractor which handles French visa applications) as well as the embassy in Sydney, because:
Residents in the UK can apply for French visas through the London embassy if they have a valid residence permit in the UK. The residence permit must be currently valid and expiring < 10 days after the date the application is given (the day you show up for your appointment in London). This will be checked by the front desk people when you first enter TLScontact, after being searched and after sitting in the front queue.
Source of this response:
I have no online source for this. I was also in a similar situation and I had no idea, and had to go and hope for the best. I was given this information by a gentleman at the front desk who checked my documents. In my case, my student visa was expiring in 10 days (although it was actually revoked by a border control officer, and I was in the country on a tourist visa-on-arrival). I was still allowed to process my application, in spite of the fact that my application was against the rules in a strict sense. This is why I encourage you to check whether they are willing to help you.
What you should do in this situation:
There is no publicly available phone number for TLScontact. You would first create your application here. You would then submit your application and then follow the link in your account to send an enquiry to TLSContact. In your enquiry you would give a brief overview and request urgently that they call you on your phone number. Several hours later, they usually comply. You would ask your question to that individual.
In parallel, you should email or call the consulate in Sydney with the same question.
2- WHO CAN APPLY FOR A VISA
Citizens from some non-EU countries are required to hold a visa when travelling to the Schengen area – to apply here, you must be resident in the UK.
I take this to mean 'ordinarily resident', for which a tourist visa (your situation) would not count. You'd need to be in the UK on a work permit or student visa. But I may be wrong on that front, and perhaps they can make exceptions, so you should call them.
Conjecture alert: I think this is because your passport is taken during your application, and if you don't have a BRP (biometric residence permit), a tourist would effectively have no identification and would be impossible for the UK to deport. I think this is why they don't want to process people who are nonresidents, but that's just my guess.