Sander takes Widge to see St. Paul's Cathedral on a Sunday after church. When they get there, Sander says, "St. Paul's...the center of things." So, according to Sander, St. Paul's Cathedral is the center of London. This could be both physical and metaphorical, of course, as St. Paul's was centered geographically and it was also where the hustle and bustle of life was happening.
Yes, you can still visit St. Paul's today, although the one standing today was actually not completed until 1710. The one that people would have seen during Shakespeare's time burned down during the Great Fire of London. Check the link below for more information.