Is this possible? Yes.

Is it something you can simply push a button in Word and have it happen? No.

This will require programming. I would recommend that instead of intercepting the Enter key, which has multiple uses in Word, that you write a macro that inserts a paragraph mark (Enter key) and inserts your time code. Then perhaps use Alt+Enter to trigger it.

If someone else hasn't done it by this evening, I'll give it a shot. I am due in court soon. It shouldn't be that difficult. However, if you are in an environment where you can't run macros, it would not be a solution. Something manual using AutoText could be done but it would make it difficult for you to do your job I'm afraid.