Definitely never a "shhh" in the middle. On those rare occasions when I go home to see my mum and then go out, that's one way in which (after 15 years absence) I can still pick out the incomers and strangers in the pub

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After being spelt Egglesham when it first appears on record in 1161 (not 1157 as often wrongly quoted) it was more or less consistently spelt Eglisham and Eglishame until the mid 1700s when most writers switched to Eaglesham or Eagleshame. However in the 1680s an "ignorant" soldier in despatches neatly summed up even the modern pronunciation by writing "Eglesum".
Except in my mum's "telephone voice" (when it ends in "ham" not "um") Arranroots has it down to a "t" (so to speak)...