The film, and I assume the book, is alternate history. It presents a fantasy version of how the play 'Hamlet' came into being.

Hamlet (full title "The Tragedy of Hamlet, Prince of Denmark") is documented by Thomas Lodge and modern scholars as being written in or before 1587. The title and the main elements of the play are based upon a much earlier work that was widely available at that time -- "The Life of Amleth". The earlier work dates to the 13th century and was in Latin; eg. Vita Amlethi by Saxo Grammaticus, a Danish historian and author.

Hamnet Shakspere is born in February of 1585. He and his fraternal twin sister were named after Hamnet Sadler, the baker who witnessed William Shakspere's will, and his wife, Judith.

So Hamnet is 2 years old and very much alive when the 200+ year old story of Hamlet is first documented as being performed in Shoreditch.

All the same, I'm a sucker for the romantic fantasy version of England in that era. 'Hamnet' may be no more based on truth than Harry Potter is but I will be buying a ticket.

The boy, Hamnet, died in 1596, aged eleven.

Four years or so later, the father wrote a play called Hamlet.

The Hamlet = Hamnet thing was made up for the fictional book and movie. In primary source documents from the era Thomas Nashe is writing about the play Hamlet in 1589 and says that he saw it two years earlier eg. 1587. Philip Henslowe hosted performances in 1594: