"Correct" is doing a lot of work there. Dialects are a thing. I have never heard anyone pronounce it like whee-ah. They would get a lot of chuckles here where it's pronounced the same as where.
Whee-ah is a little emphasis away from sounding like a donkey.
It’s not a dialect thing, the river is literally called “River Wear” irrespective of your accent or dialect. Plymouth (Plih Muth) and Tynemouth (Tyne Mouth) are pronounced completely differently and I don’t call Plymouth “Ply Mouth” just because I live near Tynemouth.
Whee-ah is a little emphasis away from sounding like a donkey.