I don't know about Germany, but in the UK the minimum wage is so low that it isn't enough to live on and the government have to make welfare benefit payments so that the people on minimum wage can survive.
That is also true in many parts of the US. In California, it is currently $8/hr, which means if you are able to work 40 hours a week (likely across two employers), you make $16,640 per year. At that income, you couldn't afford to live alone in Los Angeles, where a studio apartment would cost $1200/month, which is already 86% of your income. You could, however, qualify for subsidized housing, which would pay 80% of your rent, nutrition assistance which would give you a few hundred dollars per month to buy groceries, and recently, subsidized health insurance that would give you a basic HMO plan for free. In effect, the government (federal, state, county, and city) is basically matching the employer dollar for dollar for many minimum wage employees.
The cost of living is likely higher in the UK. Even in Los Angeles you can get a pint of beer for $4 or a six pack for $8. That is a good benchmark for minimum wages. http://www.courthousenews.com/2013/11/08/62761.htm
If that were the benchmark, Copenhagen is cheap! You can get a six-pack of Tuborg or Heineken for about $7-8, including tax, and any kind of normal local pub will have half-liters at $4-5 (though prices can go up to $10+ at fancy bars). That makes the $20/hr minimum wage look quite nice relative to costs: about three six-packs per hour of work!
However, some other things are expensive, such as housing.
That is also true in many parts of the US. In California, it is currently $8/hr, which means if you are able to work 40 hours a week (likely across two employers), you make $16,640 per year. At that income, you couldn't afford to live alone in Los Angeles, where a studio apartment would cost $1200/month, which is already 86% of your income. You could, however, qualify for subsidized housing, which would pay 80% of your rent, nutrition assistance which would give you a few hundred dollars per month to buy groceries, and recently, subsidized health insurance that would give you a basic HMO plan for free. In effect, the government (federal, state, county, and city) is basically matching the employer dollar for dollar for many minimum wage employees.