There are two broad answers here. One is that these countries adopted their own currencies during the period of American hegemony, or at least industrial and trade dominance. The other, and perhaps more important answer is decimalization.

British money retained its ancient "Lsd" system of pounds, shilling, pence (and florins, and crowns, and sovereigns, and farthings, and ha-pennies, and guinea, and groats, etc.) until the quite astoundingly late date of 1971. Commonwealth countries that wanted to adopt their own currencies typically wanted to adopt decimalization (or decimalisation I suppose) as well, and there it made sense to simply switch to a popular existing standard naming scheme of dollars and cents.

However, each switch occurred through a different route. Canada and New Zealand both adopted their own version of pounds initially. The Canadian pound was introduced in 1841 (after the act of union forming the province of canada) and was tied directly to the value of a Spanish dollar via the "Halifax rating". Since Spanish dollars were widely used in the US at the time this effectively tied the US and Canadian currencies together with a fixed exchange rate. New Zealand introduced a pound in 1840 with the effective creation of New Zealand through the Treaty of Waitangi. Since the New Zealand pound was tied to the value of silver just as the British pound was they were effectively identical currencies until New Zealand devalued theirs during the Great Depression in 1933. In 1860 the Province of Canada and other British North American colonies (which would become incorporated into the Caonfederation of Canada not long after) adopted a decimalized currency based on the dollar, in order to smooth trade relations with the US (which in 1857 ended the use of foreign coins as legal tender and established the dominance of the US dollar within the US proper).

New Zealand already had a separate coinage than the British since the 1930s and so were more free to choose their own direction with it. Finally in 1964 they decided to decimalize their currency (in 1967). There had been some discussion on possible names for the new currency but eventually they just settled on dollar.

A similar story played out in Australia, which introduced the Australian pound in 1910, devalued it in 1931, and introduced decimalization in 1966 after decades of talking about it. And they too considered many different names for their currency (there was even an official announcement it would be called the "royal" but the name was withdrawn after too much public backlash), but eventually settled on just plain old dollars.

When those countries switched to the dollar the US was a superpower dominating world trade and geopolitics, and the US dollar had become a kind of international currency well known and well used across the globe. Even if a country didn't want to signal some kind of trade, economic, or socio-cultural alignment with the US via naming its currency "dollars", the sheer familiarity of the system was a strong deciding factor.

Of course, when mother Britain decided to pursue decimalization (only very shortly after most of the major commonwealth nations had made the switch themselves) they had a greater ability to just upend the whole Lsd system and replace it with decimalized "new pounds" and "new pence" and plow right on through the confusion until everyone figured it out.