During the early 20th century, most people with some education can speak Mandarin. However, they tend to speak it with a heavy accent. What Mao speaks in his speeches isn't exactly Hunanese, but more like Mandarin with a very heavy Hunan accent. When the leaders (whom all are at least partially educated) have a meeting, they likely spoke Mandarin (many with heavy accents). By the way, Zhu De is Hakka and probably spoke Hakka natively; Zhou Enlai, from Huaian, Jiangsu, was probably the only of them who spoke a variety of Mandarin natively (yet his Mandarin is still quite different from the lingua franca, Beijing Mandarin).
You could also found recordings of Sun Yat-sen's and Chiang Kai-shek's speeches online; they both spoke with very strong accents, but they're still speaking Mandarin. The difference is that, if they speak Mandarin you'll have a hard time understanding them, but if they spoke their own dialect (Cantonese and Wu, respectively), you will likely not understand them at all.
That being said, many soldiers don't have any education and most of them couldn't speak Mandarin. That's why there was a demand for a officer corps that could speak multiple dialects.