It seems odd that these countries would abandon currencies called the "Pound" without a major deterioration in relations with the UK. Additionally, they weren't colonised by the USA.
Adding to my confusion, countries which were colonised by the USA, such as Cuba and the Philippines, didn't adopt currencies called the "Dollar".
This is actually a long story. The tl;dr is that the Pound is not a uniquely British thing and the Dollar is not an American invention.
Roman Origins of the £sd System
The modern Pound Sterling has its origins in three Roman currency denominations - librae, solidii, and denarii. The English equivalent of the three are pounds, shillings, and pence. The latter two were coins but the libra was a unit of mass used to determine the value of coins. The Roman Republic once had a currency system with the bronze *as* acting as the base unit. 1 silver denarius = 10 asses, 1 quinarius = 5 asses, 1 sestercius = 2.5 asses or 1/4 denarius. Later on in the imperial era, more coins were introduced and old coins were revalued as the purity of silver in them decreased. Diocletian for example introduced a new mostly solid gold coin called the solidus in 301 CE which was worth 1000 denarii and was defined as weighing 60 to a Roman pound or libra (328.9 g or 0.725 Imperial Pounds). However, it took 312 CE for Constantine to mint the solidus on a large scale at 72 s to a libra.
By the end of the Roman Empire, there was a large array of coins in use and various competing accounting systems. The alternative currency used was just putting bars of gold or silver on a scale and finding out how many pounds they weigh.
Reforms of Charlemagne
Europe continued to use Roman coins, often minted hundreds of years ago, long after the fall of the Western Roman Empire. Charlemagne's father Pipin the Short attempted monetary reforms to standardized the conversion rates of coins but did not succeed. In 794 Charlemagne decreed that the silver denarius, containing 1.7 grams of pure silver and weighing 1.9 grams, to be the official currency of the Holy Roman Empire. The solidus was still in circulation at the time and was valued at 12 d =1 s 240 d = 20 s = £1. This formula remained in tact in many parts of Europe until the 20th century. The solidus was extremely rare at this time as it had not been minted for two centuries but was still in circulation. In most transactions however, the s was just a unit of value and not a physical coin. The Pound was never a physical coin and was purely a way of "telling" money using a unit of weight. You can know how much silver is in a 240 d just counting how many coins there are.
This standardized system soon spread across Europe and became uniformly adopted not just in the Carolingian Empire but beyond. This wider spread was due to the Roman Catholic Church which required donations using this system, this is why King Offa of Mercia adopted the system. In the Danelaw regions of England however, the mark was the common unit of account. The mark, like the pound, was also a unit of weight and is equivalent to 234g or 8.3 ounces. It was also used to define how much coins were worth. The only coins actually in ordinary use was the denarius or penny as it was called in English.
This system of trust was eventually abused however and resulted in the debasement of currency. As the central authority of the Carolingian Empire declined, coins were no longer stamped centrally and the right to mint coins were delegated to local lords and clergy who cut corners. Over time coins also became worn and no longer weighed they were supposed to weigh. This meant coins were no longer worth what they are supposed to be worth resulting in rampant deflation. All this was exacerbated by a shortage in silver as European mines became exhausted. That was the thing about commodity money, coins were only valuable if they were made from pure valuable metals. Some mints produced purer coins than others so currency in Europe stopped being uniform.
The Almighty Thaler
To get around the problem of debasement by established mints and to save on metals, Italian city states started to mint their own larger gold or silver coins which were worth several silver pennies. Tyrol made the silver Grochen (groats), Florence and Venice made the gold florins and ducats. All these coins became widely adopted as trade currency and different countries started to make their own versions of those coins. Soon there were French groats and English groats. The Hungarian forint became one of the most widely circulated coins in Europe because Hungary was a gold mining centre. One notable coin was the Guldengrochen, guldiner for short. which was first minted in 1486, it was a silver coin of renowned purity and much bigger than other coins at the time.
The guldiner inspired another silver coin of the same size called in Bohemia called the Joachimsthaler in German or the Jáchymovský tolar in Czech. It was minted by the Count of Schlick in a valley called Joachimsthale or Jáchymov. It featured St. Joachim (the purported father of the Virgin Mary) and a lion. This coin was shortened to the thaler in German and was called the dollar in English. Like the florin, it also spawned imitators in places like Spain where it was also called the Real de a Ocho or peso (piece) and is best known in pirate lore as a piece of eight. The Spanish dollar was first minted in 1497 and was worth eight real or bits.
The Spanish Dollar soon became renowned for its regularity and purity, becoming the world's first international currency. It was widely used in the New World having spread from trade with the Spanish Main and also spread to many parts of Asia due to Spanish trade. It was the only form of Western currency China accepted. The coins were used as far as the colony of New South Wales in Australia. The coins were minted in places with huge silver deposits like Potosi in Bolivia, Lima in Peru, and Mexico City. In Spanish speaking countries these coins were known as pesos which is why so many Spanish currencies are still called pesos today. They were also a common sight in British North America due to a shortage of British pound sterling due to the policy of mercantilism which limited imports of British silver to the colonies. Shortage of British currency was a common problem in British colonies so people just used whatever coins were available. The Spanish dollar was legal tender in the US until 1857.
Adoption of the American Dollar
After American independence, the US decided to adopt the Dollar as its currency due to the excellent international reputation of the Spanish dollar and its prevalence in the Thirteen Colonies due to trade with the West Indies. Alexander Hamilton set its value as 371.25 grains (24.057 g) pure or 416 grains (26.96 g) standard silver in his Coinage Act of 1792, this was based on weighing a selection of Spanish dollar coins. The American dollar was originally pegged with the Spanish dollar. The American dollar was also the first decimal currency, being divided into units divisible by 5 or 10.
This set the trend of the dollar being associated with a decimal system while the pound sterling was associated with the old £sd system. The majority of former pound using countries renamed their currency upon adopting a decimal system. The UK (and Ireland) are notable for keeping the name after decimalization in 1971. Ireland was in a common monetary area with the UK called the sterling area until 1979 which is why they both switched to the decimal system on February 15, Decimal Day. The sterling area (officially scheduled territories) was a large union of markets across the Empire and then Commonwealth that had used either directly used the British Pound or had a local Pound pegged to the Pound Sterling. By the 1970s it was already in marked decline after a major devaluation in 1967 which is why countries started leaving the union and creating their own currencies. The area ceased to exist after 1979 due to the UK repealing all exchange controls in the Commonwealth in favour of joining the European Economic Union.
That's why when in 1857 the Province of Canada adopted a decimal system for its currency they renamed the Canadian pound to the Canadian dollar. The main reason why they made the switch was to be more compatible with close trading partner across the border. Because of Canada's closeness to the US, it and Newfoundland were the only Commonwealth countries to not join the sterling union which formed in 1931 after Britain and the Empire left the gold standard. Hong Kong was also initially a lone agent but joined at the end of WWII, their currency was called the HK Dollar even at this time though. When Australia switched to a decimal system in 1966, they similarly felt the need to differentiate themselves from the British Sterling Area which was not yet decimalized. They actually considered other names like the Royal, which was unpopular, and the Austral which they decided sounded too much like "nostril" when preceded by a number ending with "-n" (e.g., "fourteen Australs" sounds like "forty nostrils").
Sources cited
Chalmers, Robert (1893). A history of currency in the British colonies. London : Printed for H.M.S.O. by Eyre and Spottiswoode.
Hepburn, A. Barton (1903). History of coinage and currency in the United States and the perennial contest for sound money. New York: Macmillan.
Pfenningwerth, Victoria (2016). The Introduction of Decimal Currency: How We Avoided Nostrils and Learned to Love the Bill. Museum of Australian Democracy.
Porteus, John (1969). Coins in history: a survey of coinage from the reform of Diocletian to the Latin Monetary Union. London, Weidenfeld & Nicolson.
Redish, Angela (1996). "From the Carolingian Penny to the Classical Gold Standard" in Bimetalism. Cambridge University Press.
Rothbard, Murray (2002). Commodity Money in Colonial America, LewRockwell.com.