I think you're overreacting. The UK has all sorts of ties to NZ. They're in the commonwealth, they have a free trade agreement, they both part of the five eyes, etc. Both NZ and Australia are strategically important partners because of their geographical situation.
It's true that the British people are more aware of Australia but that's more to do with holidays than anything else, and relatives moving over there.
I saw reports of the cyclone and photos yesterday. In my mind it's the equivalence of floods here. Loss of property, massive disruption and sadly a small loss of life. I wouldn't think it would make more than a small mention across the world because its minor in comparison to what other countries go through, although, obviously, not for the people who are suffering.
The UK government offered assistance yesterday within hours of the cyclone hitting. I'm sure UK citizens do feel sympathy. But, just as there's nothing NZers can do about our floods, I'm not sure what you'd expect Brits to do about the cyclone.
The weather phenomenon that occur around the world, heatwaves, fires, floods, snowfall, mudslides, avalanches, are reported here pretty regularly, regardless of ties with a country. People feel sympathetic and regretful over loss of life and property. I don't know what else you expect of them.