Not St Martin-in-the-Fields, which is that huge Baroque church overlooking the eastern side of Trafalgar Square. No, the church in the nursery rhyme is St Martin Orgar, another tiny church in the City of London, just down the hill from Starbucks and round the corner from Monument station. Except that this particular church has all but vanished. Just a tower remains, now occupied by a firm of solicitors, next to a surprisingly large overgrown garden that used to be the nave. St Martin's is one of London's abandoned churches.
Pudding Lane is only a couple of streets away, so it's not surprising that this church was burnt to the ground in 1666. There were 111 churches in the City before the Great Fire, 80 of which were destroyed. St Martins was one of the unlucky 28 not to get rebuilt, being rather too close to St Clements over the road, and so the two parishes were combined. A group of Huguenots (that's old French Protestants to you and me) took over what was left of St Martins, did it up a bit and held services there until 1820. The tower was converted to become St Clements rectory in 1851, at which time St Martin's old bell was rehung in a new clock projecting out over the street. It's all a bit lost and folorn now, but I bet redevelopment of the site would net any property developer considerably more than five farthings.