I live in the south of England, Vehelpfulguy, and I know several Glaswegians who have lived here literally for decades but still sound as if they were speaking to me on Sauchiehaul Street! Perhaps the Bard retained his Warwickshire accent in a similar fashion. We just don't know what he sounded like.
Re the setting of the plays, lots of them were set in Italy, so presumably the characters shouldn't have been speaking English at all!
Below is an attempt by Andrew Gurr of Reading University to reproduce Antony's famous speech from Julius Caesar in a readable version, as he imagines Shakespeare might have spoken it.
"Frinds, Roomuns, **********, lend me yurr eerrs.
Oy coom too berry Sayzurr, nut too preyze im.
Thee eevul that men doo livz aafturr theym,
The gewd iz awft inturrid with thyr boonz.
Soo et ut bee with Sayzurr. The nerbl Brootus
Eth toowld yu Sayzurr wuz ambishius.
If it ware soo, it wuz a greevus fawlt,
Und greevusly hath Sayzurr arnsserrd it.
Heerr, undr leeve uv Brootus un the rest
-- Fur Brootus iz un onawrubl mun --
Soo aar thay ol, ol onawrubl men --
Cum Oy too speek in Sayzurrs fyoonurrul.
Hee wuz mahy frind, faythful un djust too mee,
But Brootuz sez hee wuz ambishius,
Un Brootus iz un onawrubl mun."
Doesn't sound too Brummie to me, but who knows?