There were certainly attempts to make this happen. Both New Zealand and Australia had "Acclimatisation Societies" whose goal was on one hand to fill up the "empty" wilds of the two countries with plants and animals from Europe, and to transplant any useful species from the colonies back to England. I'm going to talk mostly about New Zealand as that's where most of my knowledge lies.

Most of the traffic was from Europe to New Zealand and Australia, often with unforeseen consequences. For example, the introduction of rabbits to New Zealand significantly decreased stock capacity of land for the big run holders, resulting in bankruptcies or farmers walking off the land.

One major success for acclimatisation in New Zealand was the introduction of trout. It took several attempts back in the 1860s to successfully transport live trout ova to be hatched in New Zealand. Today trout fishing in New Zealand is an industry that brings in $88-130 million annually.

As for why not much went the other way? New Zealand didn't have long established traditions of agriculture and farming. The Maori arrived in the 1300s and several of the traditional Polynesian food crops such as taro and coconut did not survive in the colder climate.

As a result there weren't many domesticated species, and not many that would be considered 'useful'. In New Zealand the main crop was kumara, but this was quickly supplanted by the introduction of potatoes which unlike the kumara could be grown anywhere in New Zealand, not just the warmer regions. There's actually evidence that the introduction of the potato changed settlement patterns quite dramatically, especially in the cold south. Cabbages, carrots, turnips and wheat were also introduced and grown by the Maori.

Other sources of food tended to be quite labour intensive, requiring a lot of processing before they became edible. It's been suggested that cabbage trees were on the way to domestication before the Europeans arrived, but even the edible hearts of these trees needed to be baked in an earth oven for three days before they'd be palatable. There were things like karaka or tutu berries which were poisonous in their natural state. Tutu berries had to be juiced and strained to remove the poisonous seeds. Karaka berries were sometimes cultivated, but had to be steamed in an earth oven and then left under running water for several days to be made safe to eat.

Big game species like the moa were gone before Europeans arrived.

So there really just wasn't much available that would be attractive to bring back to Europe.

Further reading:
Regarding traditional use and edibility of native plants

A Field Guide to the Native Edible Plants of New Zealand by Andre Crowe

Regarding Acclimatisation

The minutes of acclimatisation meetings were usually recorded in local newspapers. Search the database at https://paperspast.natlib.govt.nz/

On the introduction and impact of potatoes

The Welcome of Strangers by Atholl Anderson