Hi,
I actually did this a long long time ago (way back somewhere in the 1980's) but instead of a wrist watch i used a digital alarm clock. The goal was to provide a 120vac output to plug something in like a record player (yeah we had them back then) when the alarm went off.
Same problem. The alarm would beep rather than stay on constant.
The solution was to use a capacitor that is charged with the alarm signal through a diode, and the circuit it triggers is made to have high input resistance. This means when the alarm sounds it charges the cap and the cap doesnt discharge very quickly so it maintains its voltage to keep the next circuit turned on.
The other problem found was that after 5 or 10 minutes the alarm would turn off, which means the cap voltage would die off and turn the circuit off. The solution there was to use a latch. It was simple enough and cheap enough so that was that. the latch means that when the alarm turns on it triggers the latch and the latch holds the logic state until a reset button is pressed.
I also like the SCR idea however, where the SCR is turned on via the gate triggered by the alarm signal. The SCR will stay on forever unless the power is removed. The reset button could then be nothing mone than a normally closed switch in series with the anode or cathode of the SCR. When the button is pressed, the SCR turns off. This of course means sometimes you'll also have to turn the alarm off though for times when you hit the reset switch soon after the SCR fires because the gate signal will again appear to turn it back on again. There are ways around this too if needed.