edit: apparently getting your money is fast and easy
The FDIC almost always pays insured depositors within a few business days of a closing, usually the next business day. Payment is made either by providing each depositor a new account at another insured institution or by issuing a check to each depositor.
The FDIC arranged for Chase to take it over. Nobody lost any money, and nobody I know had any problems with checks clearing or withdrawing cash at ATMs or paying their rent or mortgages.
I think the demise of Washington Mutual is still considered the biggest bank failure in US history, it happened while quite a lot of other bad things were happening, yet depositors ended up being totally fine. The systems in place for this are actually really good, and the FDIC is fast and competent.