When we’re talking about Zionism, we have to start with Theodor Hertzl. Hertzl was an (ethnically) Jewish reporter for the Vienna newspaper, the Neue Freie Presse, working out of their Paris Bureau. As Paris correspondent, he covered the Dreyfus Affair… The Dreyfus Affair, btw, were the events surrounding the trial of Alfred Dreyfus. Dreyfus was a Jewish army officer who was falsely accused of being a German spy. He was convicted based on perjured testimony and false evidence manufactured by a group of anti-semites in the French army.
Anyway, after covering the Dreyfus trial and the wave of anti-Semitism it set off, Hertzl came to the conclusion that fighting anti-Semitism was pointless, and that as long as Jews existed as a minority population in Europe and America, they’d always be in danger there. The only way for Jews to be safe would be to create a Jewish state. So, he, along with another journalist, Nathan Birnbaum, created the World Zionist Organization, which had its first International Congress in 1897. The World Zionist Organization then spent a lot of its time trying to convince the Ottoman Sultan to give them Palestine, and trying to get Kaiser Wilhelm of Germany to pressure the Ottoman Sultan to give them Palestine.
Meanwhile, while all this was going on in Western Europe, Jewish groups in Eastern Europe were starting their own organizations with the goal of emigration from Russia to Palestine. This became known as the First Aliyah, and lasted from around 1880 to around 1900, and the First Aliyah set up farms and communities in Palestine. The city of Rishon LeZion, for instance, dates from the First Aliyah.
I could go on if you wanted, but as you can see, Zionism started before World War II, and from the beginning, it was Palestine-centric. Herzl and the World Zionist Organization may have thought about Uganda as a backup plan, but their hearts were set on Palestine, and for the Eastern European Jewish settlers, if they weren’t going to America, Palestine tended to be their destination.
Also, you’re mistaken when you say, “they wanted to give them their own Jewish state for all of the displaced immigrants around Europe.” Britain banned most Jewish immigration to Palestine before, during, and after WWII. Most of the Jews who came during that period were smuggled in illegally. And Israeli independence only came after a three year resistance movement against Britain, and a war against the Arab nations surrounding in, in which the Arabs were backed by Britain.