I find that it takes much less time and stresses the mice much less (read: "renders mice unconscious faster") to pre-charge the container by running the CO2 into the container for a count of twenty to forty (twenty for mice with developed fur, longer for pinkies and fuzzies - it's a factor of CO2 resistance, not necessarily of animal size) before putting animals into the container.
Also, don't make the mistake of using too large a container - better to do two or three mice at a time in a fairly small container and have them go out quickly rather than trying to flood and fill a larger container with home-available equipment.
My setup is considerably more simple than what has been described.
Take one tall container - for pinkies, even a tumbler with some paper towel in the bottom works, for larger mice I use a 4X4X7 tub. Remove lid and replace with clingfilm (Saran Wrap for the Americans) - cover three quarters of the top.
Get CO2 canister, and open valve pointing output down into the container. CO2 is heavier than air, so it sinks - pushing the normal atmosphere out of the container as you fill. Count to twenty-or-forty depending on what age of mice you're using.
Close the CO2 valve but keep the cylinder handy.
Gently place mice into the container and pull the clingfilm over all but a corner of the container. Top up the CO2 - by the time you've done the clingfilm, mice with fur will have toppled over and will be deeply unconscious, mice without fur will probably be staggering around looking drunk. Keep topping up until you see the 'agonal gasps' - it looks painful, but the mice are not conscious and this is a reflex only.
Close the clingfilm over the top of the container and keep an eye on the mice.
Adult mice are usually dead by this time - my average with any mouse bigger than a fuzzy is a minute from introduction to chamber to 'being offered to the snake' (I usually pick to order on feeding days - I've got enough mice to do it, and I also alternate feeding freshkilled with feeding frozen/thawed so I know that nobody's getting picky).
Fuzzies do take longer and you may need to top up the container again if you see that you've got one 'breather' who just doesn't go as quickly as the others.
If you're working with baby-baby pinkies, you may want to consider freezing once they're in the agonal-gasp stage; this can last a while for them, and though it's not stressful for them, it may be for you.