People always mix up different immigration in different countries and airlines. It is really not that hard.
You have to enter the respective country on the relevant passport, that is usually in the statue law of the respective country.
Here is what you do:
1. Book the ticket in the Australian Passport Name.
2. When checking in at the airline counter for the flight in Australia show the Australian passport. They don't give a toss what your citizenship is, only that the name on your ticket matches.
2a. For certain countries the airline will check that you are eligible to enter the destination country (eg everyone travelling to Australia needs to have a valid visa or be a citizen). If they kick up a stink about the length of your stay in the UK show them your Hungarian passport. Box ticking exercise.
3. Go through Australian immigration and use your Australian passport since you're leaving Australia.
4. Go through UK immigration using your Hungarian passport as you should under EU law when claiming freedom of movement in another EU country. They won't give a toss that you're also Australian. If they ask (out of their own curiosity) tell them you're Aussie and point out that we always beat them when it comes to the rugby.
On the way back to Australia
1. At checkin with the airline show them your Australian passport. They will want to make sure you're allowed to go where you're going (as mentioned earlier).
2. Go through UK immigration and show them your Hungarian passport.
3. Put the Hungarian Passport away, you don't need it anymore.
4. Enter Australia on your Australian passport.
Dual citizenship is quite common and really not that big of a deal. The problem is that people don't understand that the airline and immigration are three separate mobs.
On a related note, EU citizens do not have a right of abode. They can make use of the freedom of movement rights granted to them under EU law but they will not get right of abode (a term mostly used in Hong Kong) or indefinite leave to remain (the UK equivalent). As soon as they breach the freedom of movement requirements the UK can boot them out, unless they have in the meantime obtained indefinite leave to remain.