when I googled this, it states that Blackfriars Bridge was the second bridge to be built to the City of London. On the site I visited, it states that Westminster Bridge was the second central London bridge to be built, the next crossing point being Kingston Bridge.
I have just double checked and it was Westminster Bridge (work began 1739, ended 1750), after the first bridge, London Bridge. www.fidnet.com/~dap/1955/dickens/dickens-london-map.html. Blackfriars Bridge opened in 1769, I believe.
sorry to have given a misleading answer, as I have got confused with the central London connection. The answer is definitely Kingston Bridge, which was built before Westminster Bridge and was the second bridge over the Thames.