The timer you have purchased will not work for 3-way switching.
We know this because the wires are color-coded.
And you have 2 black wires.
Good job providing that information so I can see what you have.
Both your timer wires are black, and that tells the electrician that the timer is made for single-pole only.
That being said, there are timers that have the same color wires as your 3-way switch, but the timer has to say
it works for 3-way circuit.
Usually the instruction sheet or package will say, 'for single-pole.'
Or it will say, 'for single-pole and 3-way.'
1) Aube makes programmable 7-day timers for a 3-way installation
But this timer is not for countdown.
2) I am searching the web for a 3-way countdown timers, and Levition offers one, but they require you to replace both 3-way switches and then install a 'master-and-remote' combo.
This means each timer is a different model number.
If you want help sorting out that purchase, add a comment and I will research the Levition site.
3) Smarthome sells a countdown timer for 3-way circuits
It appears to be exactly the product you are wanting.
The product has 95 ratings with 4-1/2 starshttp://www.smarthome.com/2476ST/SwitchLinc-Timer-INSTEON-Countdown-Wall-Switch-Timer-Non-Dimming-White/p.aspx
The product manual is right there on the Smarthome page.
They provide an installation video.
Smarthome is very customer friendly.
I specialize in timers and electrical devices, but I do not know the product shown above.
However I can help you install it.
Add a comment and I will help further.