Prompted by idle curiosity, I did a bit of online fossicking during my lunch break today, and it looks as though 13 Paternoster Row would have been commercial premises at the time that The Gang was supposed to have been in residence. It was one of a number of addresses out of which the publisher Charles H. Clarke operated until the early 1880s (his speciality was cheap fiction & trying to outrun his creditors). From 1890/91 onwards it was the office of James Carter, who published material in support of the Lowe brethren (a splinter group of the Plymouth Brethren). I don't know what happened in between, but the site is unlikely to have resembled Madam Vastra's domicile as shown on screen.
The Royal Bank of Scotland currently occupies 9-13 Paternoster Row, but this is evidently a recent building development, and I have no idea how far the address represents earlier house numbers.
In their amusing Facebook page, however, The Gang give their address with the modern postcode of EC4M 7EJ (it should be merely E.C. to remain in character historically). One can but wonder what sort of items the RBS post team is having to deal with in consequence.