So the voltages on the various colors are as typical for the color, and your fan really is controlled by a switched neutral. Bizarre. And since the humidistat also makes the fan run, that portion of the circuit must provide it's own switched neutral.
[With the switch turned off (or the red wire disconnected) you are reading 125V (relative to bare/white) because it is connected through the fan to the black wire; no current is flowing, so there is no voltage drop across the fan. When the switch is turned on, the red wire will read 0V (relative to bare/white) because the 125V will drop across the fan, making the fan run. You can verify this if you feel like it, but it has to be true, based on the voltage measurements so far.]
If you want to gain access to the fan junction box, you could rewire that junction box, along with the fan switch and the humidistat, to do the more normal thing of switching the hots. Then you could install an electronic timer as per normal. Otherwise, you have to work around the fact the neutral is switched in the fan switch box. You can use a mechanical timer or a relay, as discussed earlier.
Another option that should work, but feels quite wrong, is to get an electronic timer with Bare/White/Black/Red leads (I'm capitalizing to distinguish the timer leads from the wires in the switch box) and hook it up Bare to bare, White to black, Black to white, Red to red. That should still power the timer electronics properly, and should make the Red wire into a switched neutral instead of a switched hot. I don't see any way in which this is any more dangerous that your existing manual switch, but I'm a little hesitant to recommend it due to not being 100% sure I know what I don't know.
P.S. Also still unexplained is how wiring an electronic timer as usual would cause the fan to run at low speed all the time.
Cheers, Wayne