You can submit your DAT application today and they will let you know if you are eligible tomorrow. If you are eligible, you will have ~6 months to make an appointment with the testing center to take the test. After 6 months, your application will be voided and you will have to submit another application and fee if you choose to take the test. Depending on testing centers, they may or may not have an immediate available time, so it is best to look at their available time.
It is possible to get everything done in less than a week!