Welcome. My name is ***** ***** would be glad to assist.
Ok, thanks for the details.
The Intermatic timer uses the standard 3 way switch arrangement and they do not use a neutral wire for any assist on power.
The new one you have, does the same as the other , Except it requires the neutral connection for power to assist the timer operation.
If you look in the rear of the switch box, do you see any white wires joined together to each other?
Those are neutral wires and where you would add a wire and connect to the new timer.
Depending on how the residence was originally wired, there may or may not be any neutrals in the switch box.
3 ways can be installed in many ways and that is up to the original electrician.
If no white wires joined in the box, you would need to get another style timer that is labeled as "no neutral required"
Like the EJ500 Intermatic
Intermatic has several of that style , while most others all require a neutral.
Let me know any clarification needed,