It still should be your Hungarian passport.

My mother has the same dilemma. She is born Chinese with HK residency, later naturalised as an Australian. She has an Australian passport and a HKSAR passport and ID card.

Her HKSAR passport and ID card uses her maiden name, Jane Doe. Her Australian passport includes her married name, ie Jane Doe Smith.

When she travels from HK to Australia she always uses her Australian passport to avoid having to get an ETA, so I ticket her as Jane Doe Smith. She presents Australian passport at check in, even tho she enters/exits HK with ID Card.

In Australian check in she presents her Australian passport again. Although there is a visitor right to enter HK on her Australian passport she doesn't have a return ticket. But seeing her Chinese birth the agent usually skips asking for a return ticket and asks for her ID card, which is in the name of Jane Doe.

If UK check in mirrors Australian practice then on seeing your Australian birth the agent will ask for your Australian passport. It doesn't matter if the names are different.