When I applied for mine in Kent County it took almost five months. It may take a little bit depending on County. I dont know about your County but mine the gun board meet the first Tuesday of every month. I called the first Wendsday of every month and asked if it was accepted.
Man I feel your pain. My hasn't come through yet either and it's also been five weeks. I live in huntsville Alabama and every other county around us you walk in, and walk out with your license. This county you have to wait forever!!