In miles, feet, or kilometers, what size is Neverwinter from north wall to south wall and also from the west wall to the east side?
About 3k feet by 4k feet
The 5e map of Neverwinter can be found on page 52 of Sword Coast Adventurer's Guide. It unfortunately does not include a scale.
However, the 4e book Neverwinter Campaign Setting includes a map of the city by Mike Schley, which is also available on his website. Visually overlaying the two maps, they correlate well in terms of steets and buildings, so we can use the old version.
The scale on the 4e map gives Neverwinter to be about 3k feet north to south and 4k from Moonstone mark in the west to the river gate in the east.