I live very close to Eaglesham.
I have always pronounced it as: Eagles - Ham
I have ancestral relatives who married in Cambuslang in 1835 but emigrated to Australia in 1841.
This couple were married for more than 60yrs but on their diamond wedding anniversary, their very grown up children, placed an Anniversay notice in the Australian Newspaper.
They had never lived in Cambuslang, apparently had no idea how it was spelled and referred to it exactly as their would have parents said it.
It states that their parents married in Cumslang Manse