the Aus study
In case this is a common miscomprehension, I should like to point out that there is not an "Australian study". There have been many studies, in each of the legislatures that have passed MHL, by diverse research groups.
"I've pointed out a similar drop in a non MHL country"
I wasn't aware of this. The MHL countries show a pronounced drop over the course of one year, before-after laws are passed. If you have similar data for a non-MHL country I'd love to see it (as opposed to, "since cars became affordable in the UK fewer people cycle", which I can fully accept and indeed would consider a universal truth for all countries).
"How do you explain the drop in the UK?"
In a nutshell, people can now afford to drive, the infrastructure and road system has been adapted to be car-friendly and (sometimes explicitly) bike unfriendly, so they do drive.
You'd expect to see this effect everywhere, and I'm sure that the car ownership graph in every country essentially corresponds to a fall in cycling. The difference is that in years when helmet laws are passed, an extra 30-40% drops off the cycle use figures. When they are revoked, that number recovers.
It's still lower than it would be in a world without cars, and it still follows a general trend (i.e. if petrol gets expensive, more people cycle even though not as many more cycle as might have done in an environment where the risks of cycling are considered rationally).