Roche has developed an antibody test for covid-19. Per their specifications, the fastest machines can run 180 tests/hr. The total number of machines worldwide is 40,000. Thus if all testing for all other diseases were stopped and only covid testing was done, the theoretical fastest that the US and Europe could be tested is 1,000m population / 7.2m test/hr = 138 hrs = 5.7 days. There’s only 1 machine per 25,000 population so you can see they aren’t that common. This is clearly a extreme scenario since not everyone needs testing, but also not all the machines are in the US and Europe, not all of the rest of medicine has grounded to a halt, and there are collection/delivery bottlenecks. Still the antibody platform should be able to test a reasonable percentage of the population in a few weeks.
PCR testing, which detects viral RNA and thus active (or very recent) infection is significantly slower. Major medical centers like mine (which services a population of about 1 million people) can run about 100 samples a day. There are about 500 hospitals (mostly large academic centers), equipped to do this. Labcorp, Quest and Abbott together say they are ramping up to run 100,000 a day Nationwide. Thus total capacity is about to 150,000 a day, which comes to 2,000 days to test everyone in the US. Assuming we test 10% of the population to get a reasonable idea (we’ve tested about 1%), it’ll take 200 days!