Possible effective dates vary a lot. It depends on the insurance company. Each one has its own rules.
Overall, effective dates could be anytime within 1 to 90 days from when a properly completed application is delivered to the insurer.
Some carriers allow only the first of the month, or the first or the fifteenth, some any day of the month, or any day except the 28th to the end of the month, or the day after a properly completed application is received, and for some it is any date, or the first of the month, after the application is approved.
In any event, submittng an application 3 or 4 weeks in advance usually gives plenty of time for processing before the requested effective date. Medical records may be required, but not in all cases, and the underwriter may have more questions. A medical exam is usually not required, but there are a few who sometimes require it of some applicants.
If you request that the effective date is the day after the company receives the application, it may take anywhere from a day to 2 weeks or more to get notification of approval, or decline, or an adjusted offer, depending on how many applications are coming in at that time and whether or not more information or medical records are needed.
If you are in very good health, with no medical problems in the last few years, and a next day effective date is allowable, you could feel fairly confident of approval and request a next day approval date. If you do request a next day effective date and have a major medical expense before you get notification of approval, the expense would be covered provided that your application is approved. Coverage is retroactive if the application is approved. The insurance company does not know about the claim until it receives it so your application is not affected by the medical event.
However, if you request a later effective date and a new medical situation arises before the effective date, you must notify the insurer of the change so that your application information is updated.
Insurance companies do not want to issue coverage based on information that is too old. Depending on the company, applications are accepted anywhere between 45 and 90 days in advance of the requested effective date.