From Gyford: "When the Black Friars monastery was dissolved by Henry VIII in 1538, most of the buildings were left to decay, whilst some of those occupying the outer fringes of the grounds were given to people who happened to be in the King's favour at the time. One such beneficiary was Sir Thomas Carwardine who on a nod and a wink came away from the royal chamber clutching the title deeds to the priory church and east gatehouse. Having little regard for ancient buildings he promptly pulled down the church and was on the verge of doing the same with the gatehouse, but on seconds thoughts decided to make it his home."
From St Etheldreda: From about 1590 onwards the gatehouse become a building where Catholics secretly worshipped and it was repeatedly raided by government priest-hunters. Purchased by Shakespeare in 1613, though there is doubt about whether he ever lived here. The St Etheldreda website has evidence about this property that suggests Shakespeare was, late in life, a closet Catholic. Shakespeare left the property to his daughter, Susanna.
The picture source tells the story of a building which was probably this gatehouse. In 1623, 7 years after Shakespeare's death, the attic floor collapsed during a Jesuit service which was attended by 2-300 people. 90 died.