Bird call

I've tried the bird identification facility on the website, listening to the calls of all the likely birds that visit my garden, but am having no luck.  Can someone help me identify this call?  I've been hearing it throughout the Spring and Summer, many times a day.  In volume and timbre it sounds like a blackbird, but I know blackbirds don't repeat their calls.  And it's nothing like the songs of the mistle and song thrush as given on the website.

It's a rapid four-note call, repeated. The pitch is consistent, in the crack between A and A flat on the treble stave, but I will give it in A for convenience.  A; B; C sharp; A;  repeat A, B, C sharp; A.  The first two notes are like up beats, with weight given to the C sharp as if the first beat of the bar; and the C sharp and A are slurred.   Someone on another forum I posted to pointed out it is the opening of Frère Jacques - indeed it is, though I hadn't 'heard' that because it's about twice as fast as you would sing Frère Jaques.

I am in a London suburb near Wandsworth Common.  I have resident blackbirds and robins, and am visited by great tits and a few blue tits, sparrows, the occasional green finch, a secretive wren, a very exciting drop-by of a woodpecker, and goldfinch - oh and jays in the days when I put out strings of monkey nuts in shells - discontinued because a squirrel chewed the string and took the lot.  Starlings in the winter.  Wood pigeons and feral pigeons to the extent of being a nuisance.  I guess there are thrushes around but they don't alight in our small gardens.  My mystery bird seems to sing/call from a vantage point, like the blackbird, but I've never caught sight of it.

Many thanks if you can identify it!