To work as a game developer, you need the appropriate skills. For most, that includes getting a formal education. However, many game development skills can also be picked up or improved by learning by yourself.
The vast majority of the people in the games industry have a formal education, and have spent countless additional hours developing their skills in their spare time. Basically, you don’t need a formal education to work in games, but it certainly helps.
There is no formal bar you need to pass to become a game developer, so it’s impossible to say how long it takes before anyone can start working in the industry. For most however, it involves completing at least one university level education.