What are these silent H's in place names in England?

In general, word-medial /h/ tends to be lost before an unstressed vowel. Compare the pronunciations of "vehicle" and "vehicular". See https://linguistics.stackexchange.com/a/15716/5581 Some words are exceptions or have variable pronunciations, but the loss of /h/ has become standard in the place names you mention. It doesn't have to do with a British accent. However, non-British speakers may be more likely to use pronunciations with /h/ for these place names due to the influence of the spelling.

Actually, a similar phenomenon is the loss of "w" after a consonant and before an unstressed vowel, as in "answer" and "Greenwich".

I just found a related question on Linguistics SE with an answer by Colin Fine that says basically the same thing: https://linguistics.stackexchange.com/questions/8398/why-is-h-of-ham-dropped-in-english-place-names

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".