Hi timnz,
There have been many steps, actually. As you mentioned the Davis Cup and the concept of Grand Slam are closely intertwined. Without the Davis Cup, Wimbledon would likely be the World Cup of tennis and that's it. Still, there is no easy or short answer to your question.
-it is clear that from the very beginning, the US Championships weredeemed important. The US was a big country, and second only to the British Isles in terms of tennis enthusiasm. Dr Dwight organized the first 'official' lawn tennis tournament a year before the first Wimbledon (though it would probably now look like little more than a garden party).
So I think the US title had some prestige from very early. However it was not a major due to level of play. Due to geographical distance and the scarcity of international matches, it was very hard to judge their relative level. The American tennis papers debated passionately about the level of their champions relative to the British and Irish ones. In the Britons' mind there was no doubt who was superior though (and don't forget that the USA were dwarved on the world scene by Britain's power at the time, so there was a bit of condescension too, with the US seen as a 'secondary' stage for any sporting event). Dwight and Sears travelled to Britain in 1884 and had several losses, but ended up around the overall top 10/20. The first non-American to take part in the US championships was Irishman Goodbody I think.
In the 1890s things started to change, and US Champion Bob Wrenn had quite a reputation abroad. In 1895, World number 1 Joshua Pim and Harold Mahony (probably number 3) traveled to America and played a round robin in West Newton that the American players lost, but not dishonourably. Unfortunately Wrenn did not play that year. It was a shame too, that Pim and Mahony did not stay until the US championships.
So things were evolving in such a way that the US Major was more and more significant on the world stage, but it was the Davis Cup, you guessed it, that first made Newport a clear Major. Britain's losses in 1900 and 1902 (admittedly, with not their best possible team each time - no Laurie Doherty for instance !) were a real shock. The Whitman-Larned-Davis generation proved to the world that the best Americans would always be a match for any British or European player. Moreover, the result was that the USA got to defend their title at home twice... and as a consequence the best British players got to play the US Championships. So Davis Cup would soon become a yearly occasion for the best players of the challenger's to measure up against the defending country in their own tournament, as well as the DC tie.
Was Newport a full-fledged Major as it is today from the 1900s then? that is, equal to Wimbledon in prestige? Probably not, still. As you said, there was a consensus that the latter was the World championships of Lawn Tennis, a fact that was made official by the ILTF in the 1910s. Tennis was still above all a European sport, with the 3 World championships on this continent, and only pressure from the USLTA in the 1920s forced the ILTF to abrogate the World Championships titles. America thoroughly dominated tennis at that time and it seemed ludicrous to have a World Championship in Britain that seemed almost depleted when the best player was Gerald Patterson in 1922. We could say that is the moment Forest Hills really became a Major. in 1922-1925, it was clear that the Wimbledon winner was not the champion of the world, and Tilden's true challenge, apart from the Davis Cup, came only at his home tournament...
Ironically, people from yet another country wouldcontribute to confirm this new status: the changing of the guard at the top of the tennis world between Tilden and the French musketeers occured, not when they captured (and kept for 6 years) the Wimbledon title, but when they beat the Americans on their turf, in 1926.
-The Australasian followed a similar trajectory: Brookes and Wilding's tremendous success in the British/European Circuit in 1905-1907, and their capture of the Davis Cup from Great Britain in 1907 put their countries on the tennis map. While they held the Cup, challengers came to Australia and played the national tournament, which they would undoubtedly never have done if not for the Cup. American Fred Alexander won in 1908, and Irishman Jim Parke in 1912.
However, for obvious reasons of travelling difficulties, Australia never became a regular 'stop' for foreigners when it had not won the Cup, though there were some examples (most notably Borotra in 1928 ).
-the French major is a bit of a different case because of the additional 'surface' factor. The clay circuit had been around since the 1890s and some tournaments had emerged that were considered as majors for this surface: Homburg, then the South of France from 1899, Monte-Carlo from 1897, then finally the WHCC in St-Cloud from 1912 to 1923 (we could add the Olympics in 1900 and 1924 on clay).
However, were the 'Championships of Europe' or the WHCC 'majors', meaning the equals of Wimbledon or the US Championships? I don't think so. I think they were more like the Queen's or Monte-Carlo today. Prestigious enough, people talk about it in the papers and so on, but not events that really determine who is really at the top of the tennis world.
They lacked a crucial element that the French International Championships (soon to become the 'Roland Garros' event) got in 1927 when Tilden first participated: the taste of Challenge. When Big Bill won the WHCC in 1921 he was the main attraction in a field of European players. He was expected to win, but a loss would not endanger his status of World #1. In june 1927, Lacoste was the US champion, Borotra the Wimbledon Champion, Cochet the holder, Tilden was challenging them on their turf, and this event really had the feel of a 'leg' of a Tennis World Championship, which is the definition of a major, really.
-So everything was in place when the term 'Grand Slam' was used for the first time in 1933, Crawford's year. The fact that the 4 countries were the only Davis Cup winners up to this point was important, as the Australian had not had many foreign participants in the preceding years, so it could not be because of its intrinsinc level at the time.
It is interesting to note that the NY Times announced Budge had completed a 'Grand Slam' after winning Roland Garros in 1938, so the term was interpreted as 'non-calendar' Slam, unlike today.
-The history of whether this or that of the 4 majors was significant for a given year has been much discussed elsewhere on this forum. Most of the time the Australian did not really live up to its Major status, and Roland Garros did not always either. Since 1988 we have had a set of pretty much 'equal' majors, though I think a Wimbledon win would always have more prestige attached than any other one.
Jonathan