It's hard to trace the precise speed of the growth of early Christianity because there are relatively few surviving sources from its first two centuries (as is to be expected for a movement that was initially pretty small). A sociologist, Rodney Stark, has used the much better documented growth of Mormonism to suggest a model by which we might estimate the growth of early Christianity, and the predictions of the model seem to fit reasonably well with the size of the church when it becomes more visible in our sourxes in the third century. Stark's work (The Rise of Christianity) is frequently cited and, while it has a lot of issues, is generally accepted as being on the right track to suggest that the growth of the early church was comparable to the growth of Mormonism.

Islam is a more complicated case, because mohamed's followers spread as a political conquest before islam really emerged as a well-defined faith. Islam, unlike early Christianity and Mormonism, didn't start at the grass roots and work its way upward into a non-christian/non-mormon society; it was at its begining more involved with the upper levels of a new imperial bureaucracy that was trying to set itself up as an equal rival to christian byzantium. Even so, the limited evidence available suggests that the spread of islam downward to the grassroots level took about the same amount of time as the spread of christianity to the top - three or four centuries, at least, before most of the people living under islamic leaders were themselves all following their leaders' faith. This slow process of conversion was encouraged by the general lack of religious coercion against christians and jews living under muslim rule; those not working in government were generally free to keep their parents' faith, and it took many generations for islam to become the religion of (almost) everyone living in muslim lands.

I can't comment on the rise of buddhism and how it compares, i'm afraid.