Well, the expression dunny would be purely Australasian, and to a great degree unknown in England, except recognisable to some in very modern Australian humour, Dame Edna etc., and jungle reality shows.
I have never heard of any connection here between our Hyde Park and dunnies, or even Hyde Park and toilets. I cannot recall any official toilets in Hyde Park, although we do have plenty of horse traffic!
I have never heard Hyde Park in London used as a butt of jokes, except perhaps with reference to homosexual cottaging, for which it was once infamous, and may still be for all I know. Historically, 19th century etc. Hyde Park did have terrible down and out poverty stricken localities, and connections with the famous one, Rotten Row.
So I still can't throw any light on this one. For the above reasons, I assumed it referred to Hyde Park, Sydney.
Can it have any reference to Buckingham Palace, which stands nearby?