Wiring timer switch into 2-gang switch setup with common load (red) with downstream load
Don't pay attention to the colors

The red wire going to the existing switches in your box is actually a pigtail from the hot feed coming in (the junction of the two blacks), while the two black wires from the switches are the switched feeds going back out.

So, nut the black hot wire from the timer to the existing junction of always-hots that you've pulled out and a pigtail to provide always-hot to the other switch, while the appropriate black switched wire gets nutted to the red wire from the timer.  The timer neutral goes to the neutral junction (white wires) that you've pulled out, and the timer ground goes in with the rest of the grounds (which need an actual wirenut on them FYI).  Finally, the yellow 3-way wire can stay nutted/tape-insulated off as you don't need it, as you have already indicated.