I dont live in Southwell - but as a Notts person know that we acknowledge both Southwell and Suth-all.
I have always believed that those born to the area called it Suth-all and those who have 'moved-in' called it Southwell. It call it Southwell.
Added to this is, I believe, a class 'thing' i.e. what sort of money you have. If you were a middle class professional you would call it Southwell. I know 2 professional people who were not born there, but had the money to buy one of the higher price houses in the 'village' or center, that insist that it is called Southwell!!!!