How long do I have to tap "Don't send" when sending a text message from Apple Watch?
There isn't a fixed amount of time to send on Watch OS 2.0. It really depends on what you do next with the watch.

If you don't go on to other tasks, it appears to exist in a state where the "don't send" is active for over 2 minutes - well past the point where the watch screen goes dim.

I believe it's about 20 seconds once you leave the watch alone to when the message posts, but I don't get feedback and the time stamps on other devices round to the nearest minute, so getting a precise measure eluded me when testing this morning.

I just place my hand over the entire screen to blank it when I want the send to be final and leave the watch alone for a bit.