I have an update. I took the Apple Watch to the Genius Bar at my local Apple Store and they were stumped as well. Since it is still under warranty they sent it back to Apple for repair. Apple opted to replace the watch, so I was hopeful that my particular watch was defective and that the new one would work properly. I got the watch back last night and paired it with my iPhone and the same problem is happening!
Also, I purchased a second Apple Watch while the first was being repaired and the same problem is happening on it as well. So ... it must not be the watch. Now I'm wondering if it has something to do with using the watches with my iPhone SE. If anyone else has an iPhone SE paired with an Apple Watch please let me know if this issue is happening to you.
Interestingly, when I click the microphone to dictate a response to an email, everything works fine and the dictation screen stays on until I click Cancel or Done. However, when clicking the microphone to dictate a response to a text message, the screen times out after about 15 seconds and the dictation is lost.
Also, having tried everything else, I completely reset the iPhone SE and watch by erasing all data. I didn't want to restore from backup incase the issue was somehow embedded. This whole process took quite a while and I had to re-download all of my apps and setup the phone and watch again (and lost my health data). But the message dictation problem is still there. I really don't know what else to try, but it doesn't seem like very many others are having this issue, so I'm pretty baffled. I'd love to hear from an iPhone SE user with an Apple watch, otherwise, I may try pairing the watch to my wife's iPhone 6 to see if the problem may be with the iPhone SE.
I'm not really sure what else to try. I have tried all manner of turning Siri and dictation on and off, re-pairing from backup, restoring from backup, setting up as a new watch, wiping phone and watch, toggling iMessage, etc. etc. all to no avail. I'm pretty familiar with tech issues and troubleshooting, but I don't know what to make of this problem. If I find a solution, I will re-post. If anyone has any suggestions, please let me know. Not being able to dictate a response to a text message really inhibits the functionality of the watch since that is the only way to respond without using one of the canned responses. Very frustrating!