Yes, health workers will be first, for general public it will take until April (hopefully not 1st April lmao) for EU, USA, Russia, China, UK and probably few other countries. (according local news here)
Because of many potential firms releasing the vaccine in the west, there are some news that Russia and China will cooperate on manufacturing of vaccines in an attempt to produce more than their competitors.
Good thing about this is that we ordinary mortals can only expect this to result in faster production due to competition.
Since first shipments won't meet global demands it's safe to assume it will take at least until end of 2021 for the virus to completely disappear, according to WHO they are struggling to get top countries to work together for supplies, right now major countries stockpile vaccines for them self instead of working together to help poor countries.
But that's not a problem if we mortals (non doctors) really want to get it that fast, you can always become health worker in these times right?

(I guess it depends on where you live)