How long should I let my window dry out before caulking?
Cut out the old caulk now to let it all dry out, that will open everything up to the maximum to allow for drying. If it is an older home it will still dry after it is caulked. One day to dry and one day to caulk and let it set will suffice for proper adhesion. 

I suspect if the house is over 10 years old, it will not be an issue. It will still dry out. Newer homes may have a little more difficult time to dry out from the new vapor barriers out on the market.If you see drywall or plaster damage that has been around for a while, (wet is OK, water bubbles under the paint MAY be OK, black is bad) you may want to do a little removal to inspect inside the wall, just to be sure. Caulk takes a while to degrade and it does so with out notice, so check interior areas thoroughly.