There are many RT-PCR LDTs. It does not take very much to create a real-time RT-PCR LDT for COVID-19; two weeks ago the CDC posted a comprehensive description of the process on their website, but it was not a secret before then. All you need is a clinical-grade RT-PCR machine, synthesized primer/probe DNA (which many labs are equipped to do), and quality reagents.
The difference here is that the Roche instrument is pre-approved for clinical use both (1) in the US and (2) outside "high complexity" CLIA labs; the instrument is already widely deployed to hospital labs; and the instrument is highly automated compared to the basic 96-well RT-PCR machine, with a robotic stage that allows it to run thousands of samples per day with less manual work.
The WHO does not have a massive stockpile of complete test kits, and to my knowledge the countries that have scaled up testing are not relying on the WHO for anything but information. Any kits the WHO provides would need to run on an existing RT-PCR machine anyway.
CDC's test instructions and (more importantly) the target RNA/DNA sequences were posted more like 6 weeks ago -- I think I first noticed them in early February, around the time CDC announced that they had applied for an emergency use waiver from the FDA.
They've been revised considerably since then (with a big "research use only" warning on them added).
Their protocol for clinical labs is a superset of the research protocol. (Clinical labs also have extensive prior experience in this kind of protocol, and many state DPH and other labs are in continuous touch with the CDC through CDC's lab network.)
At least one company in South Korea, Seegene, has produced many tests and it is this aggressive private sector production that is largely reasonable for South Korea's effective testing. There is also a large demand for Seegene's tests around the world indicating that WHO tests are definitely not in sufficient supply.
The private sector stepping in is the best way to get enough tests out to everyone who needs them.
This is March 1st press conference from South Korea: https://news.v.daum.net/v/20200301162614985. It says Seegene accounts for 37.8% of test volume. The context is that Seegene (and only Seegene, other companies have other supply chains) had supply chain problem. As I understand problem is now resolved.
I have no clue about this, but to me it seems they just took the existing human workflow and built a robot to replicate it. Is there no more clean slate way of doing this (maybe with a lot of pipes like in a refinery?). How does one even keep one of those monsters clean?