Jonathan: Not to put words in Chaog's mouth, but I think that when he wrote "the years between World War I and World War II" he meant 1919-1939, rather than the years during each World War.
OK, sorry chaog, I misread your post.
About the pro/amateur split, as I said earlier, I don't give the pros any major before 1939. I understand that this is very unfair to Vines, Perry, Nusslein and Tilden, clearly. But this come from the lack of good tournaments involving all of these players. I'll go in some details:
-1931-1933: there is no real competition for Tilden, the pro field cannot be compared to the amateurs with Cochet, Vines, Crawford and Perry.
-1934-1936: Vines's only rival in tours is the aging Tilden. The only tournaments with a good field (ie Vines, Tilden and Nusslein at least) listed by McCauley are the French Pro 1935 and the Wembley tournaments, but Bowers and others strongly oppose the fact there were any Wembley events in 1934 or 1936. If I believe Bowers, and I acknowledge the fact that the best claycourters were amateurs, that leaves us with one good pro tournament for this period: Wembley 1935. I'm ready, on second thought to replace the weakest fastcourt amateur Major with Wembley for that year...
-1937-1938: these could have been great years! Imagine a big tournament circuit involving Perry, Vines, Tilden, Nusslein, etc. But it didn't turn out this way... I have always chosen not to count the 2-men tours as 'majors', I think it is constitutive of the game of tennis to have a minimum of 3 or 4 different opponents to win a big one. Plus, the '37 and '38 tours have not taken place in ideal conditions:both men became quickly mad at each other and demotivated.
Tournament pro tennis was totally separated, with Nusslein dominating the circuit. Only Perry played and won one tournament, the US Pro 1938, but it was terribly depleted (no Nusslein nor Tilden).
So, here is the list:
1919: Wimbledon, Forest Hills, US Clay Championships*, USA v Australia encounter**
*The World Hardcourt Championships in Paris didn't take place that year. Considering the level of play of the US players, the US Clay was then probably the best slowcourt tournament.
**In 1919 the US federation sportingly declined to play the Davis Cup, considerig that American tennis had not been nearly as hurt by the War as elsewhere. Australasia won the DC, with Patterson winning both his singles.
But after that, a friendly encounter was organized against the US. There were 6 matches, 4 singles and 2 doubles, each time with different players. The Americans didn't lose a single match. Johnston beat the old Brookes, and, in the most anticipated match, the US #2 Tilden beat Davis Cup winner Patterson, clearly proving that the US were the actual #1 team of the year. That's why I choose this encounter as a 'Major', and not the Davis Cup, to reward the actual best team. The 'winner' of the event is Tilden, as he beat the #1 player from the opposite team.
1920-1923: Wimbledon, Forest Hills, World HC Championships in Paris, Davis Cup.
1924: Wimbledon , Forest Hills, Olympics in Paris (on clay), Davis Cup
1925-1938: Wimbledon, Forest Hills, Roland Garros, Davis Cup
1939: US Pro, Wembley RR, French Pro, Southport Pro
Almost every system I have ever tried results either with Perry and Budge tied, or with Perry just narrowly ahead--not exactly the conventional wisdom. Your own research of course has them close as well, but gives Budge the edge, which would probably raise fewer eyebrows.
Budge's position depends a lot of how you deal with the war years. He was probably the best player in the world from 1937 to 1942, maybe later, I'm quite confident he could have won between 2 to 4 pro majors a year from 1940 to 1944 , had there not been a war, but, along with Von Cramm and others, he had his career cut off by historical events. He had the peak performance and the dominance to be in the GOAT shortlist, but some of his best years were taken away from him...
About Perry, even if I like him personally through what I have read, I have to recognize that he might be a bit overrated in my list, thanks to the futility of the pro tournaments circuit. For 1934-1936, Vines was at least on a par with him, and in an ideal circuit, would have split majors with Perry.
Have a nice day,
Jonathan
[CORRECTION: according to Bowers, there indeed was a 1934 Wembley tournament, only 1936 is doubtful. So 1934 and 1935 Wembley would be possible pro 'True Majors' for the 30s. Yet it seems very difficult to judge the relative levels of the pros and amateurs for these years. Vines's edge against Perry in later tours hints at a possible superiority of the pros, but the pro-am matches described by Bowers, with young Von Cramm and Henkel destroying Nusslein and Stoefen, the masters of the European pro circuit, proves it is not so simple.]