It seems, given the historical bad blood between the two peoples, that Taiwan's people would be less amenable to speaking the language of their Chinese cousins: Was this a conscious decision by government, or the source of cultural/economic pressures in the region? Or just for convenience sake due to a melting pot of dialects, similar to the problem China itself encountered?
Mandarin was the official language of the ROC and was made the official language of Taiwan after the war when the island was ceded to the ROC. Mandarin was made compulsory (and still is--Taiwanese is very rare in the school system by and large) in schools and the education of Mandarin was particularly pushed by Chen Yi, who did his best to suppress Japanese and Taiwanese--/u/keyilan talks about the attempted suppression of Japanese here. Taiwanese was, until the repeal of martial law in the 80s, illegal to use in schools and broadcast media, where only Mandarin could be spoken. Taiwanese and other local languages had already begun to decline under the Japanese, who banned the public use of all Chinese and aboriginal languages in the late 30s, after a period of gradual and pretty successful transition from Taiwanese and other languages to Japanese
Xenophon's already got this but i want to add that while mandarin may be politically dominant, that doesn't mean its the most widely spoken first language in the country. Taiwanese still boasts the largest number of speakers of it as a first language, but this will almost certainly change in the next 20 years.
The move to Mandarin in the 1940s was brutal and ultimately effective, but change is slow, and like Taiwan in the 1890s, Taiwanese is still demographically more significant than you might think. If you happen to be in Taiwan, just head south and you'll see it more clearly. Taipei has a much greater representation of Civil War era immigrants than does the rest of the island.