Here is a Tamagotchi Time Line translated from a 1997 Japanese book about Tamagotchis that I have.
1995 August - Mr.Yokoi, the president of Wiz Company came up with the idea for a virtual pet. Wiz Company does business planning for toys. He started talking to Mr.Hongo and Mr.Horimura of Bandai.
1995 November - Wiz Company made the first prototypes. (Eventually several different revision prototypes were built.) Meanwhile Mr.Hongo and Mr.Horimura had to sell the idea internally within Bandai.
1996 August - Small LCD hand-held games (such as Tetris) were very popular so timing for a product such as the Tamagotchi was felt to be very good.
1996 October 10 - Bandai started selling a test release of the Tamagotchi in certain stores. Ms.Maita* was leader of test marketing group.
1996 November 23 - Bandai released the Tamagotchi for general sale. At that time you could have bought it, but a week later it was totally sold out. High school girls loved it.
* Ms.Maita refered to here is Aki Maita, who is generally hailed as the creator of the Tamagotchi.
Note: In 1997, an IgNoble Award was presented to Akihiro Yokoi of Wiz Company in Chiba, Japan and Aki Maita of Bandai Company in Tokyo; the father and mother of Tamagotchi.