IIRC, mobility indexes crossed quite a few years ago. IOW, UK is better than the US in this respect. See, e.g., https://en.wikipedia.org/wiki/Global_Social_Mobility_Index, though it's a claim I've read based on other data and before that particular index was compiled.
Absolute economic mobility matters much more in terms of having an opportunity to get out of poverty. High economic mobility increases wage variance and therefore naturally reduces social mobility scores since the latter is a relative rank measure.
There are countries where increasing your income $10k makes you "socially mobile" and other countries where increasing your income $50k does not.
The US does poorly on this metric too: https://i.imgur.com/eYHUysQ.png [1] https://i.imgur.com/vLz5iUz.png [2]
(Note that the x-axis is "birth year.")
[1] https://equitablegrowth.org/wp-content/uploads/2021/04/04282...
[1] https://www.oecd.org/content/dam/oecd/en/publications/report...
Increasing rank is much easier than increasing income on a compressed distribution. Being able to easily increase income is much more important than being able to easily increasing rank if you are optimizing for economic opportunity.
[0] https://www.nber.org/digest/202505/wage-compression-drives-n...
That lady has some serious chops, but she said she had to leave the UK, as she was denied opportunities, because of her "Distinctive Northern Accent" (In the UK, the "North" is considered kind of "Redneck," like our South).