I'm a reception teacher, 20+ years experience, in a variety of schools. I'd say the cultural differences between white working class children and white middle class children are huge. More than between white middle class children and other cultures, and the gap is growing all the time. It's about conversation/ quality verbal interactions at home, (totally agree with pipbin), ambition and suspicion of anything that seems 'posh' (i.e academic, or difficult or elite).
So MC children come to school having been to toddler music groups, museums, ballet or gym tots type stuff, watched good TV with an adult to talk to about it, probably been to a theatre or puppet show, been swimming, had a bike or trike to ride, and had a go at at an adult's hobby (woodwork, craft, tennis, sailing, riding, sewing etc with a parent or grandparent).
I'm generalising I know, and some white WC children will have done these too, but only really in aspirational families. But lots won't have any of this cultural knowledge at all, and school life is totally based on this MC culture.