Timer Switch Problem
The good news: you have all the wires you need

The good news is that you have all the wires you'll ever need here.  The bottom cable and the leftmost top cable carry always-hot and neutral, while the center-left cable goes onward to the other light switch with its travelers and neutral.  What's more important to us, though, is that the center-right cable is a spur to the other 3-way location for the fan, with both of its travelers (red/white) and the black switched-hot, which then goes off into the far-right cable to the fan alongside neutral.

The bad news: you'll need a different timer if you want 3-way support

As you have discovered, most electronic (pushbutton) countdown timers, yours included, are built down to a price, and thus only support single pole control.  There are countdown timers of this style that support 3-way control, but they require that a matching companion switch device be installed at the other location, which just means we have to do a bit more work.

Matching the style of your original device leaves us with one option, unfortunately, though, and that's the Leviton Vizia LTB30-1LZ paired with a VP0SR-10Z remote.  This gives you the classical "pushbutton timer" functionality at the main location that you are looking for as well as ON/OFF operation at the other location.

As to the hookup

Fortunately, the LTB30-1LZ (or one of its LTBxx brethren, depending on the timeouts you want) isn't too hard to hook up, even with the remote involved.  With the power off at the breaker, of course, we start wiring this with the remote 3-way location as follows:


White from the old switch goes to the WH (neutral) terminal on the remote
Black from the old switch gets capped off with a wirenut
Red from the old switch goes to the YL/RD (traveller) terminal on the remote
And the ground from the old switch goes to the green ground screw on the remote; if the dimmer is any indication, you may have to add a ground pigtail to the existing ground bundle.


You can then button up the remote location, and move onto the timer location, which gets wired as follows:


First, the black switched-hot wire in the far-right cable (heading onward to the fan) gets tagged with (preferably red) electrical tape, as we'll need to distinguish it later on
We then move onto undoing both wirenuts of black wires, and combining all the untagged black wires with the black wire from the timer into a single wirenut
The white wire going off to the other switch location gets incorporated into the existing neutral bundle along with the white wire from the timer
The red wire going off to the other switch location gets nutted to the yellow/red wire from the timer
The tagged black switched-hot from before gets nutted to the red wire from the timer
And the existing ground bundle has the green wire from the timer, as well as a ground pigtail to the dimmer's green ground screw, incorporated into it


Once this is all wired up, you can button the box up, turn the power back on at the breaker, and enjoy your new timer!