Great response from u/VoilaVoilaWashington about Canada. I'll try to speak more about Australia.

As I understand it, India's experience with the British was caustic. The separation was relatively unfriendly, and things like the partition reflected the lack of understanding the British had. It thus made sense for India (and Pakistan/Bangladesh) to go their own way, having the only connection being through the Commonwealth, where the association is strictly voluntary, equal, and independent.

The Australian (and Canadian) experience was far less dramatic and conflicted. The UK had created the national governments, displacing the far less powerful and visible indigenous peoples and creating large immigration programs that meant the strong majorities of each population were immigrants. Those immigrants saw themselves as British, setting up new societies.

The governments created in Australia, Canada, and New Zealand were modelled on the UK. They didn't have to keep the Queen, but it made sense to do so as it was the simplest continuation of the governance model they had before, where the sovereign was represented by a colonial governor. It also reinforced the link between the new governments and the "mother government" back in London.

So, why has the monarchy survived? I'd suggest three reasons:

1. No alternative is better. Every time an alternative model is suggested, a host of potential problems arise. (This includes the fact that the UK pays all the costs of the royal family, so virtually any alternative will cost more.)
2. There is a lot of respect for the Queen. Whether that will transfer to her heirs remains to be seen, but if they follow her commitment to duty, then this is quite possible.
3. It symbolises ties between a host of nations - including the UK, Canada, New Zealand, and Australia, who tend to be extremely close and trusting. Speaking personally, I see the 3 other countries as basically brothers and trust them and wouldn't want to do anything to lessen that sentiment.

Looking specifically at Australia, there has been (and is) a republican push, resulting in a referendum in 1999. It failed comprehensively. The model put forward (basically, replacing the Queen directly with an appointed president) was seen as having the undemocratic aspects of the monarchy but none of the benefits of independence from the elected government. The alternative - an elected president - was rejected in the earlier constitutional convention because it would create a competing centre of political power, cost a lot of money, encourage populist leaders, etc.

Since 1999, the monarchy in Australia has become more and more popular. Given the failed referendum occurred only two years after the death of Princess Diana, when the popularity of the Queen/royal family/monarchy was at its lowest, you'd have to wonder whether there's really any prospect of it succeeding under more normal circumstances.

Hopefully that helps suggests why the monarchy is still in place in those 3 countries when it isn't in India.