You may not see a pH shift right away. CO2 exists in the atmosphere at about 330 to 350 ppm, and as far as I know, that's not enough to saturate water. If you introduce extra CO2, some will dissolve into water, and this extra dissolved gas drives its equilibrium w/ carbonic acid (H2CO3) to the right (assuming CO2 and H2O are on the left side of the equation). By the way, the moment the amount of dissolved CO2 starts to increase, this equilibrium will immediately shift towards producing more H2CO3.
This is precisely why rainwater is slightly acidic...it's all due to atmospheric CO2 doing its thing as a dissolved gas in the rainwater.
Coupled with your increase in dissolved gas by bubbling (in this case, CO2), you will see a drop in pH, but it may not be all that dramatic, because carbonic acid is a relatively weak acid. This is overly simple, as there is a lot of other chemistry going on in tap water (for e.g., Ca++ and Mg++ ions from our hard water getting into the act).
In other words, depending on your local water supply, you may or may not get the rapid shift in pH. Still, BeViking is correct; it is changing immediately, if not measureably using your test kit.
By the way, I've seen people use "diving bells" to allow the gaseous CO2 to have more contact time with the aquarium water. They do this by letting the bubbles of CO2 rise up into something like a small, short inverted plastic cup placed into the water. It fills w/ CO2 gas produced by your fermenter, and there is a bigger CO2 gas/water interface. In that one space in the inverted cup, if it is pure CO2, the partial pressure of CO2 is about 3,000 times greater than what it is in the atmosphere. Of course, it won't be pure CO2, because dissolved O2 and other dissolved gases will also find their way into the "diving bell" from the interface with the aquarium water, and that is only one among many posers in this scenario.
Someone I ran into at my LFS (LOL, my "real" LFS is 2-1/2 hours away) simply used a lid from a plastic food container (man, I hope he cleaned it really, really well) to hold a big bubble of CO2 from his pop bottle fermenter. He has to prune his plants weekly, and like me, he is not an expert at any of this. In fact, I got my inspiration to start fermenting from him.
Just have patience. Rome wasn't built in a day. By the way, keep checking pH. Even if nothing is happening right now, it still can get out-of-hand quickly, even with setups like ours.
v/r, N-A