I agree that Marx's analysis is fundamentally correct in describing the position of people within capitalism. Lets agree on that. It remains something worthy of everyone in the world to understand.
As well describing peoples position in relation to capitalism it is also meant to be a way of understanding oppression within capitalism. And here is where it starts to come undone, because when talking about the UK people in class categories that according to Marxist categories are oppressed may enjoy a range of material conditions to which they are politically committed in protecting.
They have become stakeholders in society, may enjoy significantly valuable assets , have material wealth putting them in the top 10-20% of earners in the world globally, may have experienced (particularly if they are older) that working what would be considered a working class job has been enough to buy them a big house, a villa, and afford to bring up kids <- using Tony Walker that east end cabby in the 7 Up series, coming of working age in the late 70s/early 80s as an example.
You could take a person leaving school now with an identical Marxist structural position as that cabby in 7 Up, get them doing the knowledge and in a cab for life and they will not be able to accrue assets and capital in the same way, their life will be precarious on a completely different scale.
Saying to someone in a precarious/no-asset/scraping rent existence that they are identical class as someone with a villa a big house in the suburbs and two well looked after kids, one who does horse riding IIRC, shows clearly the limits of saying "you are both working class". Tony voted Tory his whole life (although I think in the last episode in his 60s he went off them, forget why now)
Barbara Ehrenreich talks about how nursing was a middle class profession in the 70s/80s and has now become a working class one - the first ever nurses strike backs that change up. The difference between then and now is pay and its relation to costs of assets. I dont see how a Marxist class category on its own can account for that shift in class status <would be keen to hear an answer to that.