Between 1940-1960 there was a huge population increase, but that was only partly due to immigration from mainland China. As of 2003 only about 12% of the population were waishengren. In 2014 it was up to 14%. I don't have exact historical figures but lets assume that from 1950 it wasn't below 10%, for now, and in a minute I'll explain why that's a safe bet.

1940 population was 5,872,000, 1956 was 9,368,000, but the growth rate was still around 2.5% annually in that time, ignoring the mainlanders. ~8,504,000 is about what it would have been if that growth rate continued and there was no influx from the mainland, so we're not actually looking at that big of a jump. In fact it's just about exactly at that 10% number I proposed. That should give us a good idea of the number of people that arrived. We know about 28,000 officers arrived (this number from Davison 2003 A Short History of Taiwan) from the Mainland around 1949 but of course that wasn't the first time people were coming over.

The number, 850,000, isn't that substantial. It's really the policies they enacted that created the biggest changes.

I'm told that "many" came over and the massive influx of people helped fuel the tensions that led to the 228 incident, but it's not clear just how many came, and where specifically they came from (and what dialects did they speak).

It's a bit of an oversimplification to say this. They helped, sure, but really it's again the policies. The KMT was very heavy handed in their language policies and overnight knocked a huge number of aboriginal Taiwanese out of jobs because they could no longer use Japanese for those jobs. The upper classes were also accustomed to speaking Japanese and were not all too pleased that they suddenly had to learn a new language. By comparison to the KMT, Japan's linguistic policies were a breeze (only by comparison though; they were also oppressive policies, just enacted in a much more gradual manner).

I'd argue it's much more the heavy hand of the new government that caused 228, and that the population increase alone would not have been enough to cause that level of issues. Again, we're only talking a 10% increase. I know that that's still potentially a lot, but not anything that would necessarily leave to that degree of unrest.

A side question is whether or not many of the new waishengren arrivals were not themselves native speakers of Mandarin, but of other dialects of Chinese, and if so, which ones?

There are pockets. You'll find families who came from Cantonese or Shanghainese speaking backgrounds, for example, but now those people are primarily Mandarin speaking. The 1940s Mandarinisation policy, and the related Mandarin-only military service, really quite effectively wiped out the other non-Hoklo, non-Hakka languages in the country. In the case of Hakka and Hoklo, the Japanese policies also were pretty effective at pushing those languages into the shadows, but there was more of a sort of ethnic identity tied up there, so when in the 1990s and thereabouts policies were relaxed, and you find things like the founding of the Hakka Affairs Council, you don't see (as an example) a Cantonese association showing up. For the waishengren, they were waishengren, and were already effective with communicating in Mandarin.

Keep in mind that Mandarin as lingua franca is by no means a new things in China's history. It's long been the language that you would need to learn if you hoped to have a civil service position, Classical Chinese aside. So for the kinds of people who were able to make the move to Taiwan, they were more typically having the kind of background that would make them more likely to know Mandarin anyway.

Hope that at least helps answer your question, though I know specific demographic data would be better.

I'm also happy to offer up some reading suggestions on the linguistic shift but of course most of the literature on language policy in Taiwan is focused on Hoklo and Hakka, since languages like Cantonese and Wu were such tiny fragments in the larger demographics.