How Did Countries Introduce Nickel?

Copper nickel coins are a bit like banknotes, they have token value only. Even the Romans happily accepted orichalcum sesterces, and the sestertius was the basis of their currency system, just like the Dollar is to the U.S. currency system.
The populace is forced to have confidence in the currency system they have to use.

In personal sense are YOU happy enough to use a base metal coin that has only a tiny intrinsic value that is relative to it's publicly accepted nominal value?

The main problem is Gresham's Law. Unless the few grams of nickel was "worth" as much as the few grams of silver it was replacing, the public will cling to the intrinsically more valuable money. If the Mint re-released the 40% silver half dollar, you can bet that people would be lined up for miles to swap their change jars for real silver coins.

I don't struggle to imagine that people would have been happy to spend the new nickels, but if I went to the market, I would of course ask if the cashier had any Half Dimes before I accepted a nickel. Even a trime and a Two Cent Piece.

Going back to Rome, yes the intrinsic value of the Sestertius was largely symbolic, but not totally. When the Praetorian guard sold the empire to Didius Julianus in 193, he paid them the promised face value in reduced weight sestertii, and they killed him for it.

At one time Silver was going to be the common usage of all our coinage. However, Some guy with a White Horse and an Indian companion started using So much Silver for his bullets, this left very little for our coinage. Law enforcement people didn't want to make him mad so they all agreed to change some of our coinage to Nickel.

Back in March I Transcribed a document for Roger Burdette from 1854 that was discussing the use of nickel in the cent coin. The letter gave the price of Nickel at that time as $2 a pound. At that time copper was around 41 cents per pound.

So the new coppernickel small cent contained 4.1 grams of copper worth .37 cents and .56 grams of nickel worth .27 cents for a total metal value of .64 cents apiece. Notice that while there is only 1/8th as much nickel in the coin as copper it is worth nearly half as much.

Assuming the price of nickel stayed fairly constant, the would make the Three Cent piece 1.46 grams of copper or .13 cents and .49 grams of nickel or .21 cents for a total of .34 cents

The five cent had .34 cents in copper and .55 cents in nickel for a total of .89 cents. But the three and five cent pieces were made after the mint realized that minor coins didn't have to have nearly their full intrinsic value in them. (The intrinsic value in the bronze cent was only .26 cents.)

Most of the early introductions of nickel and cupronickel do seem to be efforts at reducing the size of base-metal coins while maintaining the "appearance of value".

Here in Australia, nickel was first contemplated for coinage in the early 1920s. The theory was that the large, bronze pennies and halfpennies would be replaced with smaller, square-shaped nickel coins (the square shape to help distinguish them from the higher-denomination silver coins). In the end, the plan was never implemented, as the costs of producing nickel coins on square planchets proved too high.

This is one factor that many countries issuing nickel coins, especially pure nickel ones, failed to take into account until it was too late. Nickel is a hard, tough metal, much more difficult to work with than the traditional coinage metals. This means production costs (labour, energy, die wear, etc) are a much higher component of the cost of making a nickel coin, and you can never get those production costs back when the coins are returned for melting. On the flip side, of course, is that nickel coins are much more long-lasting in circulation so there should be less need to return and remelt them.

Later, nickel and cupronickel alloys were introduced throughout the world to replace silver, once silver coinage became nonviable. We didn't have the same push towards removing silver from the coinage that most of the rest of the British Empire had, because Australia still had plenty of silver mines. Silver was cheap and plentiful here, and we'd have had no other use for the stuff if we'd stopped putting it in coins.

We didn't see nickel reintroduced to the coinage until decimalization in 1966, when the old predecimal silver coins were withdrawn and replaced with mostly cupronickel counterparts.

Well silver was never really available for the minor coinage other than the Half Dime, which due to its small size was very difficult to handle. And silver didn't come back into day to day commence after the war until about 1874 by which time the people had become accustomed to the more convenient sized minor coins provided by the subsidiary metal compositions. And Greshams law doesn't always hold true. Look at paper money. Under Gresham's law the idea of a preference for paper over silver (or even clad coins that have some intrinsic value) seems ridiculous. What made it work was the guaranteed conversion on demand of paper to silver. At first people didn't feel comfortable with it and that resulted in paper money passing at a discount to silver, but as time passed and the government continued to honor the guarantee the value of the paper rose until in 1874 it was passing at par with silver and the coins returned to circulation. After that the paper became the preferred form of handling large transactions because it was lighter and easier to carry and conceal. Although it had no intrinsic value in itself, the guarantee of convertibility meant that for all practical purposes paper WAS silver or gold.

One point not mentioned so far is convenience,
If you are going to have bronze coins and silver coins with intrinsic value, then you end up either with a very large bronze coin or a tiny silver one in the middle of the series - copper-nickel was useful compromise around this.