St Saviour and St Mary Overie , Southwark
The website and posters claim that St Mary Overie (short for "over the river" and not a reference to some unlikely gruesome medieval relic body part) has been the site of Christian religious worship since 606AD. Wikipedia doubts this. I have no idea but I do know there are only two naturally dry spots of land that close to the south bank of the Thames and the church is built on one. On the other are the southern workings (shortened to Southwark) of London Bridge. Actually for a brief period the southern works of one of the versions of London Bridge were not built on that dry spot and guess what, it fell down.
When you look at St Mary's, it is dominated by a very, very big central square tower. If you look at paintings, drawings and schematics of London from as far back as you like, there is that tower. For a while I could see it from my bedroom window and I used the clock and bells to tell the time instead of having a clock at home. Those were the days before the new buildings came to the Bankside area of London and so I may be one of the very last few in the 700 years of its existence to use that tower for the purpose it was built.
The architecture inside is probably best described as eclectic. Walking eastwards from the main western door there are arches you must pass through. From the south they look to be in one style and size. Coming back from the north they are quite different indeed. It actually doesn't matter for this experiment whether you are on the eastern or western sides because the four arches are all different. Given they hold that very, very big tower up in the middle of marshland, I presume that over the years architects have fiddled as much as they dared and no further.
The main part of the building has been completely rebuilt (the stone vaults on the roof are boringly regular) quite often and quite recently but as you head around the back there is an ancient sequence of chapels all joined together into one space. The roof is low and arched and supported by a thicket of very slender columns. If there were to be a quiet space for reflection this would be it, but alas you are deep in central London somewhere between the London Eye, Tate Modern, Borough Market, the City and the Tower of London. In this part of the world even the libraries are full to bursting.
Given the lack of local government in the UK until after Elizabeth I and not very much even after her (now there is lots), most civil law was run by the church and probably quite well. Lots of famous court cases have taken place in the chapels around the back. When you sit in the space you can easily see it as a court room - a number of people were sentenced to be burned to death under Mary (a fact for those who like a bit of fire in their religion) but good trials happened there too, including hearings to confirm royal appointments.
This part of London was traditionally out of the good burghers of London's control but close enough to host industries to extract their money on a daily basis. The big theatre companies of Elizabeth I's time were here, huge coaching inns and a largely regulated and moderately clean prostitution racket. The regulations, medical care and hygiene were enforced by the Bishop of Winchester (who for centuries also controlled this church - his old London house is right round the corner) but I understand the girls worked for themselves and not for the Reverend Bishop. The upshot is that this is a church, from the Bishop of Winchester's time onwards that has had a very close relationship with actors, prostitutes, play-wrights and other marginally disreputable people. So in the very back corner of the back corner is a chapel dedicated to St Andrew and those who have died of AIDS where a weekly service is held. The rest of the church is dotted with memorials to local parishioners who worked in theatre, some of whom are still quite famous five or six hundred years after they stopped working.
If you want to go to a church service, the main Sunday service can be full to standing which is, as far as I know, unique for an English cathedral. The 8am services on weekdays are a lot quieter and if you are stuck in Central London and like to go to church but Sunday is simply not going to happen, then the weekday morning services are a strongly recommended option. In general I would skip evensong as there are better choirs north of the river.