Well, it depends on the toy. If it is a video game here are a few stats that come up on Google.
Respawn Entertainment, the fall out company that formed after Infinity Ward, creators of the Call of Duty franchise, dismissed CEO Vince Zampella, retained 38 of 46 employees which resigned after Zampella was terminated. Respawn is the owner of the EA Games-published, Titanfall franchise. But this probably does not provide an accurate depiction of the number of persons or man hours needed to produce such a game.
A more specific stat comes from Leslie Benzies, president of Rockstar North Studio, a branch of Rockstar Games Studios responsible for the Grand Theft Auto series. He gave a number of 1000 people across multiple Rockstar Games Studios involved in the production of GTA 5 in an interview to http://www.develop-online.net.
As for other kinds of toys:
Hasbro, makers of American culture ( as well as many, many games we all know from our childhoods ) does not seem like the kind of place that subjects its "elves" to such harsh conditions. Deemed one of the best companies to work for in 2012, this site reports a total of about 5700 employees, over 3100 of them in the United States. http://www.toyassociation.org/ reports total sales of \$19.48 billion in the United States in 2015. Hasbro figures reached \$4.45 billion in 2015, making it top 22% of the U.S. market.
From there we can extrapolate some roughly 25,000 elves for the U.S.
Now given that the U.S. is only about 4.4% of the global population, we can further extend the estimate to over half a million elves - which seems reasonable, but we must consider also that only about 32% of the world is Christian and I have no idea how many people outside of the U.S. believe in Santa Claus, so I must assume that the low figure of approximately 180,000 elves and a high figure of 568,000 elves gives an appropriate range, the former being how many are needed if only Christian children in the world receive toys at Christmas and the later if all children in the world are given presents made by elves.
Note that the OP's figure 2+ billion children is as large as the number of people in the world who are Christian. Certainly fewer than all of those people are children, so the OP's figure must also include non-christian children. Given that, the larger end of my estimate seems reasonable.
Further more, the above figures are strictly for toys. As suggested above, including video games would be an entirely different model as a single game can involve the cooperation of up to 1000 individuals and hundreds of games are released each year on various platforms and online.