I would argue with the OP about Burwash. My lot called it Burruhsh - with an abbreviated schwa sound - can't write it.

The "ham" like meat, "uhm" with that schwa again confusion is to do with geography. The meaty hams lie in meanders, where the land looks like a pig's backside. The others were once someone's home. In Old English, the first was written "hamm", the second "ham" - I think.

It may help visitors to assume that where there's a "th" in the middle (Eltham, Streatham), the "t" belongs to the first part and is not part of a "th" sound. Elt'm, Strett'm. That way you won't get laughed at like prospective MP's or Bob Hope (who came from Eltham). On the other hand, Lewisham is not Lewis'm, it's Lewi-sham.

Then there's:
Deptford - Detf'd
Greenwich - Grenitch
Peckham - Peck'm
Dulwich - Dullitch
Wrotham - Root'm
Trottiscliffe - Trosley
Shipbourne - Shibburn
Meopham - you should be able to work out that the p belongs to the beginning, not an f sound. Mep'm.
Eynsford - Aynsf'd

And I never got to bottom of Cirencester, which has multiple other versions. Cicester, Cister, Ciceter, Ciren (almost Zoiren) - which seem to have class distinctions. The first three go with an upper accent, the last more rural. I decided, when my parents lived over there, to stick to the full name to avoid problems.

A lot of variations are to do with omitting bits of names - economical speech, versions like Jo'burg for Johannesburg. And that habit has been round for ages. It is known that in late Roman Britain, Rochester - Durobrivae - was pronounced Robri, which became Robrichester, and so what it is now. Something similar happened with York, once Eboracum.

Back to Will Fly's Sussex, I've a dialect book written by an upper middle class woman with a condescending attitude to those further down in society who tells how the doctor really couldn't understand that the local who wanted him to go to an emergency in I Urstood was referring to High Hurstwood.

My grandfather came from Lambrurst (Lamberhurst), and his sister lived in Wodurst (Wadhurst). And there was Crowbruh - Crowborough.