Short answers, pulled from the story.
The first prisoner executed at the Church of St Peter ad Vincula was Ranulf Flambard, a Norman bishop. He was incarcerated on the 15th of August 1100 by King Henry I, setting a grim precedent for the chapel.
The current edifice of the Church of St Peter ad Vincula dates from 1519 to 1520. This reconstruction was undertaken by Sir Richard Cholmondeley, the Lieutenant of the Tower, after a fire destroyed the previous church in 1512.
Queen Anne Boleyn, Queen Catherine Howard, and Lady Jane Grey are buried in the Church of St Peter ad Vincula. Anne Boleyn was executed on the 19th of May 1536 and buried in the chapel, though the exact location of her grave has been lost to history.
During the reign of Henry III, the church housed an enclosed cell for an anchorite, a hermit who lived a life of seclusion. Henry III supported the living expenses of at least three different recluses, including Brother William, Idonee de Boclaund, and Geoffrey le Hermit.
Sir Richard Cholmondeley likely designed the current Church of St Peter ad Vincula with the help of William Vertue. Cholmondeley, whose tomb now rests within the nave, oversaw the rebuilding after the fire of 1512.
The canonry of the Church of St Peter ad Vincula was reinstated in 2012 when Roger Hall was installed as the first canon in three centuries. The canonry had been abolished in 1685 before this modern restoration of the role.