(I am reading "excepted" as "accepted". My apologies if this was not your intent)
Minimum wage is more complicated in its economic effects than that. And it certainly is not universally accepted as a success story by economists, although it may be accepted as inevitable by politicians.
It tends to inflate the price of services, and shove manufacturing jobs overseas. This is because of the old economist's saying "you can't get your haircut in china" - a short hand for the fact that cutting hair, serving coffee and making beds in a hotel is not a function you can outsource to china. However, the higher you set the NMW, the more you incentivise producers of moveable products to offshore the production.
The NMW in Britain is set at quite a low level, so it has limited economic effect. I doubt you will find any subject on which economists all agree, but I can absolutely guarantee that price control (which NMW is a form of) is certainly not such a subject.
I'm not saying its right or wrong. As I said, at the current level, I think its pretty irrelevant.