Traditionally, mass produced injection molded parts (with a lifetime part count of 100,000 and above) typically have a 3 month turn from design release to the plastic part manufacturer to the time when first shots are coming off the molds and are ready for first article inspection (FAI), if your manufacturers are US based.
As of 2019, in China, if your design group is working closely with your manufacturer, you can probably get it done in 4-6 weeks. China is an order of magnitude faster than US when it comes to bringing mass produced parts to market.
If you want to keep manufacturing in the US, (and there are many reasons to do so in the beginning of your product development cycle - not to mention IP production reasons), there are services like Protomold that takes less time and incurs a lower tooling cost by creating a more rudimentary mold with less cooling that will run slower and wear out sooner. The lead times for these services are typically shortened to the 6-8 week range.