The film, which will have a one-week awards-qualifying run in the US before the end of this year, before being released more widely in 2019, stars Branagh as Shakespeare and Dench as his wife, Anne Hathaway.
The film reportedly examines their by-then troubled marriage, and their grief at the death of their only son, Hamnet.
Shakespeare died three years later, aged 52; Hathaway in 1623, aged 67.
The plot involves Shakespeare’s return to Stratford-upon-Avon in 1613 after a devastating fire destroys the Globe theatre.