NAAT is done locally inside the pharmacy but only one or may be two qualified persons can do it. If that person is out then your luck is also out because your sample would have to wait for the person's return to get done, together with all other samples that got piled up with yours.
Therefore you should make several appointments at different stores. For critical time frame, make MULTIPLE preparation, if you want assurance but do not want to pay an arm and a leg, and you should not either because the tests are Free by government rules. It even says for travel purpose is eligible.
I don't know what good a result from sample taken 5 days ago can be of any use.