Less than you might think, since spread requires movement between
population centers, such as by trade/commerce.
Case in point, Cortez brought smallpox to Mexico in 1519, which
contributed to his victory there ... but its spread to North or South
America was quite limited by the indigenous population: it was
repeated contacts from Europeans in new locations that did it.
Case in point, the defeat of the Inca Empire is popularly attributed to
Pizarro in 1532, but one needs to first understand the huge role that
smallpox had and how it got there. Best guess is that it was another
outside introduction, by Pascual de Andagoya's military conquest failure
in 1522. It took two years for Smallpox to reach to Quito to kill the Inca
king Huayna Capac and his successor Ninan Cuyochi in 1524 (as well
as reportedly a ~third of the population). Their deaths triggered a civil war
among two younger sons (and another ~third died in that civil war) before
Huáscar lost to Atahualpa ... and it was only then, after the plague and war
that Pizarro and his men had their conquest in 1532 where Pizarro had the
luck to capture Atahualpa on what was Atahualpa's homeward victory march.