YANBU.
Having lived in London and also elsewhere around the country, you are only being realistic that living in London is SO much more expensive. I have lived in some of the other fairly expensive cities (Edinburgh, bristol...) but costs wise they are nothing on London.
In London a 1 bed flat will be at least £1000 per month and that is in the cheaper areas. Compare that to the northern town where I come from, where you can rent a 4 bed house for £500. Yes people can move out of London if they can't afford to live there, however we would then all be stuffed as we would not have anyone to do the necessary but low paid jobs in London.
I don't understand how anyone can live on NMW in London. Hats off to anyone who can. NMW makes life tough wherever you are in the country but especially in London - costs really are so much higher and it's not entitled to say so.
The only thing that I do wonder though is that would pretty much everyone on NMW be getting housing benefit? Presumably they can't afford to ever buy on that wage so renting it is, and when renting on that income I think you would be given HB? If so then they effectively do have a top up to their NMW as the HB they receive is higher and goes some way to addressing the gap in accommodation costs.
There is a "London living wage" which some companies use to reflect the fact you can't really live on NMW in London. I think a living wage should be paid every where, of course, but the truth is you can get by with slightly less difficulty on that income in other parts of the country. I won't say it is easy anywhere!