I quote from Ralph Merrifield's "Roman London" (Cassell, London, 1969, pp. 95-7):
On the southern side of the ancient [Roman] road, and therefore beneath the middle of the present street, just to the east of Cannon Street Station, there stood in the Middle Ages a mysterious stone monument which, at least as early as the twelfth centruy, was known simply as London Stone [footnote to an 1189 reference]. ...
In 1742 the worn stump, which had been for some time protected by a stone cover, was removed to the north side of the street, and in 1798 it was set in a stone case in the south external wall of St. Swithin's Church. The church was destroyed by bombing in the war and the Stone was removed when the ruined wall was demolished in 1961. It was found to be merely the rounded apex of the monument, made of Clipsham Limestone and shaped rather like a tea-cosy. ... the anceint fetish stone of London ...
... Since it is not a natural monolith, but a monument shaped by masons from quarried stone, it is unlikely that it is a pagan Saxon sacred stone ... It was certainly not a wayside cross of te Christian Saxon period, since in that case some memory would have been preserved in its name. ... Its Roman origin has been doubted because it is of Clipsham Stone, an oolitic limestone which was extensively used for building in East Anglia in the Middle Ages. We do not kow that this stone was exploited to any extent during the Roman period ...
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