It is our policy, and I'm almost 99% sure FAA interpretation, that until the crew is powering up for take off, you would still need to add a take off alternate if the weather went below landing mins at your departure airport.
We add/change destination alternates, take-off altnernates, ETOPS Alternates, certain MEL's (if the flight crew can accomplish the procedure), route changes, etc before take off but after block out as required. Usually, if the actuals are anwyhere close i'll add the t/o atlternates early, doesn't cost you anything (usually fuel isn't an issue for a take off altn) and it beats waiting till the last minute. If it has to be last minute, the plane waits until they have one to go. Not ideal, but it's the rules.
Personally, if they take off and poop an engine or have a fire, emergency is going to be declared and all that paper goes out the window, then all the rules are thrown out except for the safety of flight one.