Not all the "H"s that appear to be silent are supposed to be silent, and whilst it is a common speech habit in London, you'll hear it in Manchester too, for example "'urry up, we'll miss't bus to 'rrogate" (Hurry up or we'll miss our bus to Harrogate).
I was brought up in Lewisham, South London and my Mum would implore us not to "drop your H's" and roll her eyes in despair when we did, because she wanted us to grow up "speaking properly". Other letters are dropped rather casually, Lewisham Hospital was inevitably "Lew-shum Osp-itall".
Lewisham is the borough directly south of Greenwich that I can assure you is referred to as "Grin-idge" by those who live in the area. Anyone using "Green-wich" marks themselves out as a non-Londoner.
It is not only leading "H"s that get dropped. The "g" in the "ing" of words is commonly dropped, as are some "t"s. For example: Tooting Bec would be "Too-in Bec", Tottenham would be "Tot-nam". The "th" in North and South are routinely pronounced with a soft "f" sound. Consequently, I was born in "Saarf Landon" but rarely went "nawf ov the riv-ahh" (Thames).
But don't despair, sometimes, what you see is what you get. The City of London street known as Cheapside is pronounced just as you see it, "cheap-side", however, for years, as a child my Dad played a trick and convinced me it was pronounced "Khee-ap-siddy".