What level of mathematics, science and overall cognitive ability is necessary to be a good private pilot - say up to commercial license?
You'd really have to look at the CASA requirements to work out exactly what you need. As far as I know, being able to pass their subjects is sufficient.
I didn't go through the various stages of licensing, so I'm no expert on it. I had a restricted private licence, and I jumped from there to an ATPL...with the military covering the bits in the middle.
Off the top of my head though, at the private level, common sense will be the greatest asset you can have. The maths is pretty simple, basically being a bit of trig, and the ability to count to 60. Once you get to commercial though, you'll need to start understanding the 'why', so at that point the educational needs will increase.
A lot of what I was taught allows me to understand why things happen. I haven't been taught how to handle every situation under the sun, but the background allows me to work out my own solutions for most things. They generally don't have to be exact either...
Here's a question? You're going to a point that's exactly 500 miles due east of your current location. There's another place that's 100 miles north of the half way point on your track. So, if you were to fly to destination, and then directly to that diversion point, you'd fly a total of 500+269 = 769 miles.
You only have fuel to allow you to fly 600 miles. How far along track can you go before you have to divert?
Easy. Okay, now we'll put in a north easterly wind at 50 knots. Your TAS is 200 knots.
Work it out using a chart and a piece of thread.