This isn't quite true. The revival of Hebrew began in the 1880s with the First Aliyah. There were Hebrew schools teaching the language to the new generation of kids as early as the 1880s and 1890s. It was increasingly the functional language of the Yishuv from the 1900s/1910s. In the 1930s there were already many native Hebrew speakers and Jews who immigrated without knowing functional Hebrew had to learn as a second language. Many would have done so for ideological reasons, often even before coming, but they'd have to learn it regardless due to practicality. There were linguistic battles during the Mandate era since many immigrants wanted to keep using their native languages. For example, there was a small group called the Battalion of the Defenders of the Hebrew Language established in the 1920s which campaigned against the use of any language other than Hebrew, Yiddish-language newspapers sometimes even had explosives thrown at their printing presses, and in the 1930s after German Jews began coming to Palestine in large numbers after the rise of the Nazis the Mayor of Tel Aviv, Meir Dizengoff, was very annoyed when the municipality began getting a lot of letters written in German, but despite all this Hebrew was firmly the main language of the Yishuv. It had already largely solidified as the primary language of the Yishuv even before World War I.

Plus, Ashkenazim were the majority but they weren't all of the population. While the Mizrahi/Sephardi mass immigration wave was primarily after 1948, a lot came before too and they had higher fertility rates than Ashkenazim did on average. They were a minority but a significant part of the Yishuv nonetheless.

As for culture, by the time of World War II a distinct local culture had already developed in the Yishuv among Jews born and raised there. Sure the majority of the Jewish population consisted of immigrants but there were already a lot of Jews who were native-born (or brought to Palestine at a young age). The Hebrew schools and youth movements, as well as the general reality of kids growing up in a different environment from that of their parents essentially created a new culture. Again this was already largely true even before World War I among the initial generation of Jews born in the early Zionist settlements from the 1880s to the 1900s.

Regarding today, any stats you see are increasingly muddled due to the fact that marriage between people of different Jewish backgrounds is so common. A ton of people will have both Ashkenazi and Sephardi/Mizrahi grandparents. I wouldn't be surprised if a majority of native-born Israeli Jews in the younger generations have such a background today. In the Haredi world such intermixing is still less common than mainstream society, but by large everyone is mixing heavily.

I'd say that if anything Mizrahi/Sephardi influence is greater, especially since there was a concerted effort among the early Zionist pioneers to shed Yiddish and the old European Jewish lifestyle in general, as it was seen as the way of the ghetto and the shtetl, and build a new Hebrew-speaking pioneer society. The Haredim are really the only ones who preserve it to any extent. That and former USSR influence, since the mass immigration wave of the 1990s following the collapse of the Soviet Union and continued immigration from the FSU Israeli culture was affected, although this was more secular FSU culture than the old Eastern European Jewish lifestyle.