This map was created by the consultancy company 24 Acoustics in 2012:

The green area shows how far the bells would have been audible in 1851 assuming a south-west wind (which is the most common prevailing wind in London). The smaller blue area is the audibility today, thanks to the greater ambient noise from vehicles and other machinery, and to higher buildings blocking the sound.

The sound would have travelled as far as Waltham Forest and Newham to the east, and covered the whole City, but did not penetrate into the West End at all.

The bells of the Church of St Mary-le-Bow are first mentioned in the year 1469, though the description implies that they already existed at that date. The church was destroyed in the Great Fire of London in 1666, and new bells were made in 1677. They were recast several times since as they became damaged or worn. The bells were destroyed again in 1941 by German bombers; new ones were made (using some of the metal salvaged from the old bells) in 1956.

There are now twelve bells, ranging in size from the smallest at 285 kg, to the large tenor bell (the ‘Great Bell of Bow’ from the nursery rhyme) at 2.14 tons.

I was unable to find a specific reference to how loud they are, but typically, church bells are up to 120 decibels if you’re standing right next to them, and 75 decibels in the ringing room at the base of the church tower.