If you are not approved, your check will be refunded. If your agent can do so, then you will be insured even before the formal policy is issued. But you should read your policy to see whether it provides a grace period, and if so, how long it runs. Even if your policy does not include a grace period, the law of your state may give you one. You can find out by contacting your state department of insurance.
Most states have laws requiring that insurance companies pay claims within a certain time period.
The general time frame for payment is 30 to 60 days, though it will vary from state to state. In such a situation the insurance company could be liable to you for the amount you were owed under the policy plus a penalty for emotional distress and possibly even punitive damages. Practically speaking, however, you would have to initiate a lawsuit to collect damages for late payment of a claim. That could consume a lot of time and involve considerable hassle. To determine what your state's regulations are, you can contact the state's department of insurance or an attorney.