In the US, a police check is very quick. At least in the state of Missouri. Give them your ID, they run a check, and print out your report and give it to you. Takes, maybe 5 minutes. But that is only for the city. For the state is a fingerprint process, and is same day as well…if you do it in person. And of course, the FBI thing.
Unfortunately, TECRO in the US would only except an FBI report since police and state reports are only local. This caused a problem since the KCTECRO had a local woman working in it who kept saying the local report was OK, but the Taiwanese woman was not saying that, and getting really impatient with everyone. I swear, embassies and consulates the world over are all the same.