1-2 days usually. I waited a week once to see if my son's baseball schedule would come out. I didn't want him to miss the first game. It didnt. So I RSVP'd yes to the birthday party and sure enough the schedule came out a few days later, with a conflict.
My child is only 6, if that matters, but I try to make sure he gets to every birthday party he's invited to. He's social. He likes almost everyone. I'd still encourage him to attend a party for a child he wasn't crazy about.
I've done several parties and it seems about 1/4 respond right away and the rest wait until last minute. Doesn't seem to matter whether it's paper invitations or evite.