Baseball is popular in Finland, it has been among the three or four most popular team sports here since the 1920s.
The thing is, though, that this is not American baseball, but
Finnish baseball that was created as a separate sport by one Lauri Pihkala after WWI, based on American baseball but also traditional Finnish ball games. Being somewhat acquainted with Finnish baseball (we all play it at school, and I was for a long time a loyal supporter of a local top division team), I consider the Finnish version superior to the original because the process of pitching and hitting a ball follows a different pattern and allows more tactical options for both sides, especially the one that is "in" and thus IMO makes the game more interesting.
The reason Pihkala introduced the game to Finland was explicitly to improve the physical condition of the Finnish youth (both men and women, the women's
pesäpallo leagues have always been some of the most prestigious and well-attended women's team sports in Finland) - and to improve the military skills of the men, such as running, upper body strength and throwing hand granades. Pihkala was a nationalist and a member of the Civil Guard (conservative volunteer militia) movement, and through his influence, sports and physical exercise became an important feature of the Civil Guards training in the interwar period and also allowed pesäpallo to spread and grow as a sport.
To apply this to other European nations: perhaps in Britain and France similar figures as Pihkala introduce baseball as a sport to improve the condition of the young people - and also military skills. Perhaps use such organisations as the scout movement and popularise the sport through there. In Germany, have the NSDAP adopt the sport, too, with overt connotations to do with military prowess.
If the sport manages to catch on in the 20s, then we could see international competitions in the 30s that would raise the sport's prestige in Europe. Having it as a exhibition sport at the 1928 Olympics could lead to it becoming official Olympic sport by 1936 - and, say, seeing a US vs. Nazi Germany baseball final (where the Germans would hold their own by finally fall to the American side) at the Berlin Olympics might give the sport a real boost in Europe.
During and after the war, have the presence of US troops in Europe bolster the growth of the sport, with, say, exhibition events between US military teams in occupied Germany, where to the chagrin of US commanders, the bloody locals might often beat the American teams due to (by now) a significant local baseball tradition. The chance to actually win the Americans time to time would help the sport to grow also here, not only confining it mainly to the US expat community - leading to a baseball Bundesliga founded in Western Germany in the 50s (like IOTL) and becoming a popular fixture of German sports in the 60s (unlike IOTL). Allow similar development in other bigger and smaller Western European nations, especially within NATO, and by the late 1950s you could have quite lively Baseball World Cups. Finally by the 1960s or 1970s, also other nations beside the US could realistically expect to win the Cup.
If, through the Scout movement, say, the sport was also brought to Eastern Europe in the 20s and 30s, say in Poland and Czechoslovakia, baseball might survive there also after the war, though Moscow and the Communist officials might take it into their sights as "Western, bourgeois influence". And so as baseball grows in Western Europe, in the Communist bloc it would be seen as a form of rebellion to have posters of American, Western German or British baseball stars and secretly playing games when the officials are not looking.
After the USSR falls, then the former Communist bloc countries would flock to set up baseball teams to join "the West", considered something that would be part and parcel with going to NATO...
