This is all rather NSW-specific, stuff I found out for a friend who was researching teaching options.

Secondary teachers do get to choose their subject: you have to choose when you start study to become a teacher. So you get into the uni course and you have to tell them at enrolment what subject(s) you are going to specialise in.

Note that if you want to do one of the conversion courses (ie DipEd/BTeach, where you have an existing degree and you do a shorter course to switch to teaching) you have to have done relevant courses inside your other degree. For example, here's what they say you need to have to convert to teach Biology in NSW:

'A major in science (3 years of degree level study with at least 4 units at level
2 or above) of 4 units in biology and 2 units in one of chemistry or physics." (A "unit" means a university subject, and "level" is which university year you took it in.)'

You'd have to check with a university which of your nursing subjects count.

The NSW info is at NSW Institute of Teachers: Subject Content Requirements (abridged)