I'm sorry; I need some clarification here. There's about 300 years between the crucifixion and Constantine.

In what way do you mean that was fast? Or are you talking about something else entirely?

Michele Salzman has a book The Making of a Christian Aristocracy that has some stats on religious persuasion in the late empire, but it's been heavily criticized for basically bad math. I don't know if it helps with your question about speed though.

EDIT: Mathiesen is the critic. In addition to being a historian, he worked for NASA building a space station. His review of Salzman says she don't know how to math. They go back and forth over something like three years on that, more than 60 pages, which I read.

I gotta say, I'm sorry, but if you're arguing about math, I'll take the NASA guy over the historian.