The tests have to be sent to a lab and it depends in some cases were the team is located in relation to the lab with respect to how long until they get back results. 5 labs are used across the country with the major lab in New Jersey so for most part if the test is given on a Friday morning for instance at best the results might be late Friday night but in some cases possibly not till next morning. Teams like Jets and Giants as example would be able to get their test results back a little quicker the most teams.
Here is portion of an article from a Saints beat writer on their testing process:
As it stands right now, NFL players, coaches and staffers test most days of the week, but they don't routinely test on game day — which is part of the reason behind why the Tennessee Titans' outbreak has concerned those in NFL circles.
For a normal game week, players, coaches and staffers get tested Monday-Saturday in the morning. They test on their Tuesday off-day as well. Those tests are typical nasal swabs — not the 6-inch nasopharyngeal swabs that were used around the world in the early days of the pandemic.
Those tests — for the Saints, specifically — get driven via courier to the BioReference Laboratories facility in Houston, which is a 348-mile drive.
There are several BioReference facilities serving the NFL's swath of tests from its 32 teams, including one site in New Jersey that had an "isolated contamination" issue with several tests back on Aug. 22.
Typically, the Saints get their test results back in the early hours of the morning.
It is probable that the Saints received their Saturday test results back from the Houston BioReference lab a little earlier than normal, which is why the news regarding Burton's positive test dropped just before 11 p.m. CT.
https://www.nola.com/sports/saints/article_263350e0-0605-11eb-9fb9-8fd53d8bd868.html