"Everybody Knows Me in My Old Brown Hat" sung by Harry Champion, can be found on a CD called Cockney Bill of London Town, Windyridge CDR3.
I have also found references to a song called "I See You've Got Your Old Brown Hat On."
Copied from http://www.soft.net.uk/creative/chorusese.htm :
EVERYBODY KNOWS ME IN MY OLD BROWN HAT
Everybody knows me by my old brown hat
That I've got upon my pim-pim-pimple.
It only cost me a tanner down the cut
But it looks all right on my little wooden nut.
Oh I say, what do you think of that?