Depends, given the CDR in his build, as well as his internal mechanic where landing Q shaves some time off his other cooldowns. In general it's pretty short; Ezreal typically maxes his E second, so you can judge the cooldown based on his lvl and stuff, but I think generally a rule of thumb is you want to act within a couple seconds if possible. I think a lot of Ezreal players are very arrogant and play over-aggressive, so you'll see them E into fights all the time, which is when you jump on them.
I think if the Ezreal you're playing is overly cautious, the best thing you can do is not let yourself get poked down, which either means you super hard engage on the enemy and just wombo the crap out of his team, so even if he escapes, his team is dead and he can't really do much on his own except annoy you. Or just avoid prolonged stand-offs. You know that time in the game where everyone just groups mid and sits there doing nothing? Yea, that's where he pokes you to death. Instead of that, move around the map and go for objectives, make Ezreal come to you, and he'll probably make mistakes which you can capitalize on (watch your smites in case of steals from his ult). Also, don't just adopt one strategy, use both, or others.
I think blue Ezreal also falls off late game, since he doesn't really rely on crit, just his damage from Q and Muramana (though with ER, he probably has a bit of crit. His Q doesn't crit, though, so he'll have to get into auto range to utilize it, making him much more vulnerable).