Try about a year. There are currently more than 200,000 ahead of you. Most of these are requests for WWII campaign/service medals that were never issued. Each request is investigated by means of a variety of secondary sources to determine what medals each individual is entitled to. This alone involves a good deal of time and the employees - especially the younger, newer ones - don't get in a hurry, which adds to the time element.
A copy of his discharge papers will be on file with the county clerk ONLY if he filed same. This was NOT mandatory. Individuals did so for various reasons, the most common being to establish eligibility for bonuses paid by counties and/or states.
If a copy is, in fact, on file with the Monroe County Clerk, send me a copy of your copy and I'll "interpret" it for you.