From VoiceOfUnreason's Dona Es Religion (sp?) thread in the War Academy:
Spontaneous Conversion
At the end of each player's turn, each of his cities is checked against each religion to see if the city will spontaneously convert to that religion. To be eligible, the city must be free of religions, and must be Trade Network Connected to the holy city.
Very rarely, you will see the announcement that you have founded a religion, and immediately find that it has spread to other cities. This demonstrates the timing of the game mechanics; after you hit end turn, your research occurs, followed shortly thereafter by CvCity::doReligion. The technology splash screen appears after all of this, at the beginning of your subsequent turn. With truly unfortunate timing (and rather dubious prioritization), you could theoretically find that - having founded a religion - you have a converted city (immediate spread from the holy city) but ruins where your holy city may have briefly stood (were it razed by one of the opposing players.
Trade Network connected means almost what it sounds like; there must be a connection between the converted city and the holy city - that is, the cities must be part of the same plot group. Plot groups are determined by explicit routes (roads, railroads), and terrain routes (oceans, rivers, coastline). These latter types depend on which technologies have been discovered by the owner(s) of the cities. Why almost? Because cities can connect through a closed border.
Two factors determine the probability that the city will convert, given that it is connected to the holy city.
The first factor is plot distance, expressed as a percentage of the maximum possible plot distance on the map.
Two different distance terms are used in CIV. Step distance treats diagonals as one unit of distance. Plot distance treats them as roughly 1.5. When you watch a city expand its cultural borders, you are seeing it expand using plot distance
In an unmodded game, you'll observe about 10% per turn if you are in the immediate neighborhood of the holy city, falling off to 0.1% at the furthest corners of the map.
The second factor is the amount of influence the holy city has - the only building which affects the influence is the shrine, which doubles the probability that the religion will spread.
Although all of the religions are checked each turn, the test is interrupted if the city converts, so you will only see one spontaneous conversion in a city. The religions are checked in order, so there's a probably-not-perceptible bias in favor of Judaism.