If you look at a 16 day queen calendar, the cells are capped at day 9, the queen emerges at day 16, give or take 12 hours.
Most books suggest pulling the queen cells on Day 14, mainly for reason of "candling" or knowing at that time whether the queen is fully developed, and thus not wasting time placing them, or using resources for dud cells.
I normally pull mine on day 13, allowing me to have a bit of flexibility with the extra day if I'm really busy. (and once a queen comes out early and kills the rest, you also know why....
Day 11 is the hump period. This is the day that some suggest it's OK to move a cell, due to the fact that the queen naturally becomes dislodged from the royal jelly plug. Prior to this day, any bumping or jarring will mean a dead queen (or a severely deformed queen) if she is dislodge from the royal jelly.
I would certainly wait for day 12 or after, before attempting to cut out and move cells from natural comb. Moving a queen cell from a grafting bar, is much easier than cutting out a cell embedded in wax.
As a last resort, usually within 24 hours of the queen emerging, the bees will start removing the wax from the end of the cell. this allows the queen to more easily chew her way out. Not that I'm suggesting you wait to see this, but more for the advice that if you see this happening, get the cells out quick. Otherwise you will have queen wars, and lose what you have worked for.
BTW...the above comment about the bees removing wax, along with concerns of a lack of "communications with the cells having bees in contact with the cell, and heat issues, is why I do not "cage" queen cells, especially with the push in type wire cages. You can always capture a few bees under the wire, but I wonder if it's always enough or the right type of bees. With all the ways to raise cells and transfer them into nucs, etc., I find caging queen cells with wire cages to be about the worst option one can choose.