A great many paper products (e.g., most ordinary copy paper) is essentially acid-free because the manufacturers add calcium carbonate to neutralize the acidity of the wood pulp that is their raw material. They do this not to keep museum people happy, but to protect their machinery from the long-term effects of the acidic lignin in wood fiber.
Many products soid as “acid free” have a neutral or even alkaline pH, BUT they still contain lignin which, over time, will yield acidic decomposition products. Only those psper products that are both acid-free AND lignin-free should be regarded as being safe for long-term archival use. (Some acid-free paper products that are “buffered” with added calcium carbonate will counteract the acid-generating decompsition of lignin — for a time, but not forever.
I can’t imagine any paper product going from neutral to acidic in a matter of minutes, but the rate of such a change would depend upon a number of factors including the species of wood in the pulp and the original pH of the product, as well as the temperature, humidity and chemical composition of the environment.
Note also that perspiration, skin oils and some cosmetic products may be mildly acidic. So if any of those things got onto acid-free paper, and a pH test pen was then used on the spot, it might register an acidic reading on a spot that had been neutral only minutes previously.
Bottom line — go for acid-free AND lignin-free paper products, but expect them to be significantly more costly.