Easy script for making and overlaying video counters?
In that case Adobe After Effects can do this easily, which probably also produced the effect in the video.
It's actually the only software where I know you can do this without too much manual labor. Other products usually only offer a timer effect which would be tedious to use for these kind of things.

Though if you want to do this based on just a single key press and don't want to set keyframes manually you will have to write a script for After Effects, this can be done either via Adobe ExtendScript which is a subset of JavaScript or alternatively Windows Batch and Apple Script. I'd suggest using ExtendScript, it has better documentation and is relativley easy to use.

Be advised that this won't be a task that you can finish in 1-2 days if you haven't worked with After Effects yet. Especially the scripting part. Do it manually first and then look how you can automate it via a script.