I'm glad this got picked up, I read about the controversy somewhere else, not the BBC, and I honestly thought it was a prank.
The article I read [sorry I didn't save the refs] didn't call it white supremacist, I think it took the line that it was classist because only privileged women have bare nails.. or something.
The words 'Foulcault-addled grievance gerbils' sprang to mind when I realised it was not a prank.
I did a quick google and it looks like having fancy nails done costs around €40-60 around here.
If you have €40-60 to spare, after you've covered all the necessary outgoings like food and rent and heating and travel costs and school books and pocket money for the kids etc. etc., go for it.
But be upfront and honest and admit that having €40-60 to spare after all those everyday expenses actually makes you privileged. Many people would need every cent of that just to keep food on the table, a roof over the heads and the light and heat on, and they have bare nails because they are not privileged enough to afford fancy ones.