I can try to address Canada's situation re: Falklands War.

It should be noted that the 1931 Statute of Westminster established that the (at the time) Dominions of Canada, Australia and New Zealand would have legislative independence from the UK. This meant that these dominions became self-governing countries and had independent legal authority. Thus, they had the full power to pass laws without Britain. Britain still handled defense and foreign affairs matters to some degree prior to the 1931 statute.

In the past, due to its long historical and cultural ties to Britain, Canada automatically joined Britain in various conflicts out of loyalty and imperial duty to king and country. The Boer War and WWI immediately come to mind, as well as WWII -- except here, Canada declared war on Germany independently of Britain, as a sign of its growing autonomy from the Empire.

The Falklands War was one situation where Canada found itself in a bind, politically and economically. It agreed to impose import and export sanctions on Argentina and encouraged negotiations towards a peaceful outcome, but declined to cancel a contract to send nuclear fuel to the Argentines for a Canadian-built nuclear reactor. The government was concerned that, had it cancelled the contract, Argentina may be inclined not to use the reactor for peaceful purposes.

Canada also refused to declare support for British claims to sovereignty over the Falklands, arguing that while Britain was within its rights to retake the islands as an act of self-defense, the matter of sovereignty should be left to international arbitration. This perceived neutrality irritated both the Thatcher UK government and the Americans, as President Reagan had voiced his support for the British position and offered U.S. military help if asked. He was also a staunch ally of UK PM Thatcher.

Canadian PM Pierre Trudeau was elected as a progressive nationalist who was less inclined to conduct foreign affairs within the sphere of Cold War-era politics. He had advocated for a "Third Option" Canadian diplomatic policy that was more independent of, and occasionally at odds with, the U.S. and its traditional allies. Trudeau was also lukewarm about the Commonwealth and didn't like Thatcher or Reagan personally.

While the Canadian government didn't rule out providing military aid to Britain if it was requested in theory, it was non-committal and did not expect such a request to be likely.