If you apply for Medicare during the regular IEP time and do it in the 3-month timeframe before you turn 65, your coverage will start on the first day of the month of your birthday. For example, if your birthday is June 3rd, your IEP is March to May. If you enroll in any of those months, then your coverage will begin on June 1st. If you wait until June to enroll, coverage begins the following month. However, if you enroll in any of the last three months of your IEP, it will take two to three months for coverage to begin after the time you enroll (not after your birthday month). So, for the June 3rd birthday, if you were to enroll in August, you would not get coverage until October or November.
There is one exception to the timeline above. If your birthday falls on the first day of the month, then everything gets shifted back a month. For example, if your birthday is on June 1st, your IEP is from February to August. If you enroll during the February to April window, then coverage begins on May 1st. When enrolling during the month of your birthday (June), coverage begins July 1st.