It varies. Towns don't "cash checks". The PD has to turn over the check to the town treasurer, with documentation as to the source and purpose of said payment. Town treasurer has to deposit said check to the town's account with attribution as to source and purpose of said deposit.
Some towns do it quickly, others wait days/weeks/months before they do all the paperwork and deposit them.
Date of cashing/depositing a check mean absolutely nothing in gov't terms.
BTW, it is an "application fee" . . . no refunds if you are denied (not likely unless you are a legally DQ'd person).