Why exactly would inflation increase?.
You also seem to be forgetting that we get inflation, regardless as to if we have a minimum wage, you do know inflation exists globally in nations without NMW laws?.
It's expanding the money supply which is a greater cause of inflation (which I may add, most workers haven't had a rise to cover them for)
You don't need to increase prices of goods as a result of increased wages at the bottom, look at the total income distribution statistics in the UK & you will see quite clearly a minor reduction in wages at the very top could easily pay for an increase in the bottom.
To base cost to create a item, or provide a service is a combination of a number of factors, wages are only one of them - what I'm suggesting is changing the distribution of those wages to even out the playing field (taking a few percent off the top & putting it on the bottom).
A company which can't afford to pay it's staff enough to live on, should go bust & it's customer go to a company with a more balanced wage system.
That's capitalism yo, the state covering the wages is the biggest case of (corporate) welfare I can think of.
You make it sound like most businesses can't afford to increase the wages of the bottom earners by a few grand a year - rubbish, any business that can't handle a minor increase in wages (which they would benefit from by having a increase in sales due to the people of the UK having an increased disposable income) doesn't deserve to be in business in the UK.