Welcome to the forum @Appleby
I was born in Howard Row, which was next to Clifton Row. Howard Row was demolished shortly after my birth in 1947 but Clifton Row remained throughout my childhood and I played there often with classmates in the 50s. There were Applebys in Clifton Row then but I'm afraid I can't remember anything about them. I attended the colliery school which had 2 classes. There was a boy named Appleby there but not in my class, so he would be a year or so older/younger than me. I think his name was James.
Alan recommended that you look in the gallery among the photos in the album Netherton/Nedderton old photos. I'm going to suggest that you also look in another album " Netherton/Nedderton old photos 2. There you'll find quite a few class photos for both the Netherton school (infants) and the Nedderton school (juniors) where all colliery kids were educated. With a bit of luck your friend might be able to recognise some family facial feature that could help to identify an Appleby.
In that album you'll also find 2 of my favourite photos: Clifton Row, back (facing the outside 'netties' )and the more presentable Clifton Row, front facing the gardens.