And you're talking about Australia, and the OP about the UK, so it's the same: different for each country, so not a matter of one "global common sense" nor "one rule for all". There will be people with both Korean and Thai citizenships, also, so what I say is relevant.
Here even is usual that the family names differ, as if the mother is Korean, the Korean family name would normally be listed first in the children's passport, while the father's family name would be listed first on the other country's passport. So, you can be Amy Lee on the Korean one and Angelica Amy Brown on the American one. It's very common here.
Not a matter of common sense. I mean, it's totally country dependant. In Korea (my home country) many people have two passports (I mean, dual citizenship) with two different legal names, and it's never been an issue. Don't know for the UK, but for the US and other countries, having two different names associated to two different citizenships has never been an issue here.