“Dr Swan believes that the rapidly decreasing fertility rate means that most men will be unable to produce viable sperm by 2045.”
For everyone else:
[In 2027, after 18 years of total human infertility and global depression, the world is on the brink of collapse and humanity faces extinction.](https://en.wikipedia.org/wiki/Children_of_Men#Plot)
So according to the movie it started in 2009. The movie was released in 2006.