Can a new save withdraw from the pokemon bank?

I can say 100% that yes, you can. The only restriction is that you must have your pokedex. I have tried this, and can confirm. I know the pokedex thing because I attempted to access the bank with a different game in, and that is specifically what Bridget says.

On top of that, the bank is only tied to your Nintendo ID. I have traded between three different games with the bank already, two Ys and an X. So that is 100% possible, as well.

Keep in mind you can't use this to transfer items. If your Pokemon is holding an item, it will be sent back to your bag, and if the bag is full it will be deleted instead.