Apple does not actually replace the battery in your iPod. For the nano, the way the battery replacement program works is they just give you a factory refurbished iPod nano in exchange for yours (and the normal battery replacement fee of $59 in the USA).
Since yours was damaged by water damage, they will likely only offer you an out of warranty replacement, for $79 USD, but for that you will get a refurbished nano, which has a new battery and will appear like new. Since you really cannot tell if it is just the battery, or the charging circuits or who knows what, that is a better solution anyway, it seems to me.
However - beware that you are responsible for all content on the nano. Apple does not guarantee your data and they offer no service to transfer it. So make sure you have everything backed up so you can load it back onto your replacement nano when you get it home.
P.S. you can check pricing and everything under the support page for iPod. Also there will be instructions if you want to ship it in instead of taking it to an Apple Store in person.
http://www.apple.com/support/ipod/