All Hallows By The Tower
One of the oldest churches in London, All Hallows by the Tower (that's the Tower of London, btw) has a Saxon arch built with recycled Roman tiles which dates back to AD 725. As you'd expect, its sheer longevity means it has an illustrious history; Thomas More is buried here, founder of Pennsylvania William Penn was baptised here in 1644, and the doctor, missionary, theologian, Bach scholar and organist Albert Schweitzer made several recordings of Bach's organ music here. The church was heavily damaged by bombing in WWII but has been extensively restored. It's still a working church, and hosts regular organ recitals.
The nearest Tube Tower Hill, and the church is served by the 15 and N15 bus routes.