What recourse does an employer have for employees who do not turn in their timesheets on time? You can discipline the employees in the same manner you would anytime an employee fails or refuses to comply with company policy. What you cannot do is hold their pay "hostage" in retribution for failing to turn in their timesheet.
You have to pay the employee if you're aware he or she worked, even if you have to guestimate their hours. Of course as an evil-genuius HR/payroll friend of mine has suggested, "There's nothing to prevent the company from estimating LOW."
By the way, "salaried" is merely a pay method. You need to be clear on which jobs are exempt and which are non-exempt in order to make sure your payroll practices are complying with the law.
If the company shows they're serious about turning timesheets in on time by giving disciplinary actions to the worst offenders, I expect you'll find the problem largely clears up. In all fairness though, you should first tell all employees that henceforth they will be subject to formal disciplinary action if they turn in their timesheets late.
A person, who is nice to you, but rude to a waiter, is not a nice person. (This is very important. Pay attention. It never fails.)