Except that the Old Bailey is a court building, not a church, and has no bells. The church referred to in the nursery rhyme is the one just across the road, the oddly-named St Sepulchre without Newgate (don't worry, it's a Crusades thing). You wouldn't guess from looking but this is the largest parish church in the City of London, described by Sir John Betjeman as "high, wide and handsome", and the tower contains a peal of twelve bells. Henry Wood, founder of the Promenade concerts, learnt to play the organ here and his ashes now lie in the Musicians' Chapel. The church is also the last resting place of Captain JohnSmith, unpaid star of the Disney cartoon Pocohontas, and one-time Governor of Virginia.
A glass case inside the church contains the handbell which used to be rung to wake condemned prisoners at Newgate Prison on execution mornings. This prison held those awaiting trial at the neighbouring Old Bailey, which has been London's most important criminal court since medieval times. In 1834 the court's jurisdiction spread to cover the most serious cases from the whole of the South East, including Oscar Wilde's infamous sodomy trial. One hundred years ago Newgate Prison and the old Old Bailey were demolished to make way for the current Central Criminal Court, judging the fate of evildoers including Dr Crippen, the Kray twins and Jeffrey Archer.
An absolutely brilliant website charts the history of the Old Bailey, including full details of all the trials there between 1714 and 1799. For example, 250 years ago this month there were 62 trials, mostly for theft, with many of the convicted subsequently transported to America. I wonder if any of your forefathers appear in the records.