Thankfully, it's a pretty simple override to just bridge the two sides of the switch. It seems like there's a timer IC that counts how long to keep the light on for, and "holding the button down" just continuously resets that timer.
This Sharp atomic alarm clock features a nice backlight that shuts off after 5 seconds. My mother-in-law, however, would prefer the light stay on so she can see it in the middle of the night. In this video, we'll open up the Sharp atomic alarm clock, override the backlight switch, and install a DC converter so that she doesn't have to worry about batteries running out in the middle of the night!