This is the parish marker of St Mary-le-Strand in TQ3080 : Strand Lane WC2 it is attached to the wall of "The Playground". This was part of King's College School that moved to Wimbledon in 1908. However tradition dies hard, the new King's College bicycle sheds are still referred to as being "in the playground" by KCL staff.
1888 puts it in the William Ewart Gladstone era; but is it he? If not, who then? There are apocryphal stories of notable Conservatives confusing the NLC for a public convenience when on their way to the House. Perhaps this is one of 'them' leaving.