Before trying to answer this question, it's worth considering what kinds of assumptions are baked into it.

Assumption 1) The features that exist in English dialects from the UK today have been relatively stable across time.

This isn't true for any dialect, not one from the UK, not one from the US, and not any dialect that exists in any language anywhere in the world. Languages change, and the features that we might think of as major identifying traits of a dialect may not have existed at some point in the past. The American English that is spoken today isn't identical to the English spoken in North America three centuries ago, and the British English spoken in the UK isn't the same either. We can't simply compare the dialects as they exist today and assume that American English was at some point identical to British English and then lost the features that we associate with British English. Likewise, you imagine Australia and New Zealand "adapting" to a British dialect, which again positions British English as unchanging while colonial Englishes are the ones that change, adapt, or lose features.

You didn't volunteer any examples, but I'll use one because it will also help me complicate the other assumptions here. One feature that is common to many dialects in England, Australia, and New Zealand is a tendency to drop r sounds in positions that are after vowels and before consonants. This tendency is often accompanied by a tendency to insert r sounds in situations where one word ends with a vowel and the next word begins with one, such that law and order starts to sound more like lore and order and the idea of becomes the idear of. We could call this a feature of British English that exists today, and your question would frame this as a feature that has been "lost" in American English, but it's actually the opposite. It's a feature that wasn't widespread at all in the early colonial period and that became much more common in the late 18th and early 19th centuries.

Assumption 2) Dialect areas are fairly homogeneous.

They aren't. There are people whose families have been living in North America for generations that you would likely swear up and down aren't American or Canadian speakers of English, like the speakers of the Ocracoke brogue, whose speech is often described by other Americans as sounding like a thick Australian accent (but it's definitely not). Likewise, if you went to the UK, I can almost guarantee you that you could find people who didn't have any particular feature that you could name as identifying British English. For instance, there are several rhotic dialects used throughout the UK (especially in Scotland), and at the same time there are also non-rhotic dialects in the US (in places like Boston that often maintained closer relationships to the UK even after American independence). This can have important consequences because immigration doesn't draw equally on speakers of every dialect. Nova Scotia and Newfoundland dialects, for instance, developed as a result of heavy immigration from Scotland and Ireland, respectively.

Assumption 3) We are good at perceiving differences in dialect

We aren't. We are good at perceiving that there is a difference, but people typically aren't very good at articulating what those differences are or understanding how they relate to other dialects. Your suggestion that Scottish English is roughly similar to London English especially gives the game away here. Nobody in the UK would ever dream of making that claim, and it is often a running joke that Scottish accents are hard to understand for people in England. When you hear a Scottish accent, you know it's definitely not American, and so your brain ends up grouping it together with English accents because you hear both sets of accents in movies filmed in the UK. Even though they are much more geographically close together than most American accents, they couldn't be more different from one another.

But don't feel too bad about it, because you're not alone. We are all really bad at understanding our perceptions of dialects and what they signify. We tend to rely overly much on certain iconic shibboleths of the dialects (like non-rhotic dialects) and gloss over major differences that more clearly separate the dialects in a way that the speakers readily recognize, even if we don't do so as outsiders. Likewise, we are often really bad at interpreting the social meaning of dialects.

With those assumptions out of the way, let's tackle your question!

The biggest difference in the North American experience versus the Australian and New Zealand experiences is the timeframe in which colonization was taking place. Australian colonization begins in 1788 with the claiming of New South Wales, but it would really take several more decades before large numbers of colonists would arrive. Colonization in New Zealand begins in 1840 with the signing of the Treaty of Waitangi. By contrast, the US at this point in time is not only thoroughly colonized but has also won the Revolutionary War. The British English that informed the development of American English was, depending on how you look at it, decades or possibly even a century older than the British English that informed the development of Australian and New Zealand English.

But Canada I think provides a useful guide for thinking about your question because of its relationship that continued with the UK after the US Revolutionary War. Many of the earliest speakers of English in Canada did not actually come from the UK at all. They were Americans who had fought on England's side during the Revolutionary War and went to Canada, and already by the mid 19th century, visitors to Canada from the UK perceived a difference in their speech. A Scottish immigrant to Canada, Reverend Archibald Constable Geikie coined the term "Canadian English" in 1857 and described it as full of "lawless and vulgar innovations" and stated that it "inevitably partakes of the same influences" as American English (qted. from Stefan Dollinger and Sandra Clarke, "On the autonomy and homogeneity of Canadian English").

The Reverend Geikie is just as much a victim of poor assumptions about language as people are today, and it's hard to disentangle his claims from a context in which he would inevitably see difference as deficiency. The superiority of British English was taken for granted, and American and Canadian dialects were seen as having fallen from grandeur, in much the same way that you imagine the dialects as having lost British traits, when the truth was that what Geikie knew as British English was a construct developed from prescriptive norms, and that both sets of dialects had developed over time and in sometimes divergent ways (but also sometimes in convergent ways due to influence upon one another).

In any event, my point is that already in 1857, both American and Canadian English are being perceived by immigrants from the UK as being markedly different from British English. At the same time, the colonial projects in Australia and New Zealand are in their relative infancy, with most speakers of English having immigrated from England within their lifetimes.