How long is the cool down or wait to add members to a gym that has just been flipped?

Around 10 minutes (2 spins) from my experience...

If you helped then there is no wait...but you need to be in battle when the gym falls.

If you battled the last battle, you must be picking your Pokémon in the gym to defend it before the first person adds his/her Pokémon in. Or else you have to way around 4–7 minutes.

If you didn't battle or a Pokémon got kicked out, you have to wait for 10 minutes.

If you both place defenders in synch with each other no wait. I've said this many times on this board and I'm surprised such a protip hasn't become widespread common knowledge. I've done this dozens of times with two people and a handful with three