Bubbles, I think your background is law, which is by definition a pretty English profession. (And from past MN threads a pretty old fashioned and snobbish one.)
DH works in a specialist mid-level city role. It's really not about whether YSJ or York. I think only one of his current immediate colleagues is British anyway. British colleagues he has worked with have had degrees from places like Huddersfield or Southampton Solent, had work experience in local backroom operations of national/international banks then worked their way up including a transfer to London. Equally there are plenty in his organisation who grew up relatively locally, started at the bottom and proved themselves. DH went to Oxford, took a Masters at a RG and has a fairly traditional route through. In his field you are recruited on your performance in your last job rather than your University.
Xenia will be able to confirm that in many London private schools, white British are a minority. In part because parents able to afford fees tend to reflect the diversity found in many City workplaces, but also because many international parents and British ethnic minorities find the idea of sending your children away to school, very strange.
It depends a lot on subject. The likelihood is that degrees from York will be more rigorous reflecting the higher entry requirements. For those subjects/professions where what you cover at University (either technical knowledge or the ability to research evidence and argue) is important, York will PROBABLY be your better bet. But not always. I do not know much about YSJ, though a relative recently did a foundation course there, having mucked up her A levels and moved onto the degree of her choice elsewhere, but vocational degrees with good placements can be a good choice. (Oxbridge is not the best for costume design, product design or music technology, yet I know people making good livings in all.)
Its about subject, course content, and fit.