I run it where the moment the ambusher decides to attack, that initiates the roll. There are plenty of class abilities that nullify the disadvantage of being surprised, so it seems unfair to grant the ambusher a "free" attack given this is a class ability (I'm talking about advantage on initiative rolls). Call it a sixth sense or whatever, but it really just means they suddenly become aware that something is trying to attack them. If the source is hidden, they can use their action to try to detect the source of danger (perception vs. stealth roll). Other options include running for cover while warning the party. Plenty of ways to narrate the quick reactions and keen senses of a character.