I really don't think it would work, where would the money come from?
Here's some hypothetical numbers.
If the minimum wage was £10 per hour in Newcastle and you ran a corner shop, to pay your staff, (never mind your bills) lets say you had to put a 100% markup on every item. So if you buy your mars bars in for 50p you'd sell them for £1, lettuce for 75p you'd sell them for £1.50.
If the minimum wage was £20 per hour in London and you ran a corner shop, to pay your staff you would have to either sell double the quantity or increase the markup on your products. To only need to sell the same quantities of products your mars bars will have to be at least £1.50. (My figures might not be accurate, but I hope you see my point and I have massively simplified this.)
Unfortunately this will be a minimum as your electricity bills, gas, rates etc will all be higher as the (local) suppliers will have higher wage bills too and many many costs will spiral out of control. And your mars bars you buy in to sell in your London corner shop will start off as higher than 50p as your local supplier will have higher wage bills than a Newcastle supplier. But then, if your Newcastle corner shop has to buy things in from London, his base price will be higher than it would from a local supplier. It gets more and more complicated and spirals out of control again.
Also, in places like Tesco etc. if the minimum wage is £10 per hour in Newcastle and £20 per hour in London then as a woman living in Newcastle I could be getting paid half the wage a man living in London could be getting paid. Would screw up the chance for equal pay for women, depending on how many men and women worked in each area.
Also, how would you sort out which areas had which minimum wage? As the town I used to live in in the NE basically had one 'posh' expensive side and a cheaper side. The posher side would need a higher minimum wage than where I lived, otherwise everyone would sell up and come live near me. (Apart from those already on more than the minimum wage.) But Newcastle is so much more expensive, so would they have a higher minimum wage than my old town? But some people living in Newcastle worked in my old town, would they get the Newcastle minimum wage or my old town minimum wage? Would my old town subsidise the Newcastle dwellers? Or the other way around, some people living in my old town work in Newcastle. Would they get a higher Newcastle minimum wage and be better off than their neighbour doing the same job in my old town? Or would the Newcastle job be able to pay them less as they live out of town, therefore Newcastle employers would be more likely to employ people from out of town, but they'd need more money to be able to afford public transport or petrol and parking charges. It is very, very complicated
My brain is going to explode, I also tend to over complicate things anyway.
But here's something controversial, wouldn't it be easier to charge exactly the same amount for something where ever it is sold, north or south, and pay everyone the same wage, no matter what job they do? Then we would all be equal and live in a Communist country! (I am no way a Communist.)
It would probably be fairer to force the costs of living down in the more expensive areas, particularly London, rather than trying to bring in graded minimum wages.
(Sorry everyone, I might be looking at things incorrectly, so if I am please ignore me rather than ripping me apart and derailing the thread. If I am right then hopefully I have made a good, even if oversimplified, point.)