Income and wealth inequality are greater in the US, but perhaps there are more privileges that money can buy in the UK so that inequality is more meaningful there.
I may have misunderstood, as this doesn't appear to make sense.
The NHS provides a baseline level of care that generally (but with some variance across different conditions) provides clinical outcomes on a part with other comparable industrialised nations. It is possible, if one wishes, to buy what is effectively a top-up policy that will shave a little time off here and there, or give more comfortable accommodation, but there is no evidence I am aware of that suggests this has a substantive impact on the clinical outcome itself.
In that sense this is not an inequality any more than, say, my deciding to buy a Mercedes when my neighbours have Fords or Vauxhalls. A Ford is just as capable as getting me from A to B as my E Class, but I prefer the comfort of the latter. My neighbours may instead prefer an extra foreign holiday every year, or be buying a second home out in the country.
The difference, I would suggest, between us and the US is that in the UK (and most other socialised systems) there is universal coverage. if I walk into a US A&E department with no insurance and (say) cancer, they will stabilise me but not provide the chemo.
If I walk into my local hospital in the same circumstances then as long as I am a permanent UK resident who has been in the country for more than 3 months they will give me full treatment - cradle to grave, as it were. No co-pay/deductible/excess. No bankrupcy if I go above an insurance ceiling.
There is undobtedly inequality in the UK; more so than the Scandanavian countries, for example. But everyone is legally entitled to free education of a broadly comparable standard (there are some differences between the home nations), health treatment, and has access to what is sometimes suggested to be a fairly generous (especially if you're looking at it from Calais) welfare net that includes such things as housing benefit, disability benefit, and income support. If rich nobby people want to send their bairns to a posh school for £10k per term then more so the fool them, but it doesn't automatically suggest some sort of Victorian class divide.