Tim thinks that the Low Pay Commission is coming round to his way of thinking. He notes a report in Telegraph and says:

We said that if you bring a min wage, one which continually rises above general wage inflation, then you will get to a point where it does severely crimp employment prospects. And it will be first evident among the young, untrained and untried.

Now, let's leave aside the fact that the Commission has not done anything as vulgar as actually present new facts here; a chat to a Torygraph reporter does not count as evidence. And let's also leave aside the fact that others have found the link between the NMW and youth unemployment to be surprisingly elusive.
Let's instead do some simple maths. There are 4.06 million 18-24 year-olds not in full-time education. 2.77 million of these are in work, and 1.29m - 31.8% - are unemployed or inactive. This means that to get the employment rate up to 80% for this group would require an increase in employment of 17.3%.
How could such a rise be achieved by abolishing the NMW? There are only two possibilities: either the abolition allows wages to fall a very long way; or the price elasticity of demand for young workers is very high.
The former seems unlikely. I suspect that even if people did turn up to work for derisory pay, the effort they put in would be minimal. Remember, it's not just laws that set the minimum wage - social norms do too. And it's possible that these put a floor under wages that isn't far below the legal one.
The latter is also doubtful. High elasticities usually exist only where there are close substitutes. Yes, it's likely that an average 20 year-old is a close substitute for an average 21-year-old, and this will generate a high price-elasticity of demand for 20-year-olds. To this extent, abolishing the youth NMW might expand youth employment a lot.
But the corollary of this is that the youth NMW is currently having the opposite effect - of supporting demand for 21-year-olds.
What's more, this is only true for averages, and we're not dealing with averages here. As Tim says, the youth NMW bears heaviest upon the least skilled, the 'untrained and untried.' But these, by definition, have fewer close substitutes and so the price elasticity of demand for them is likely to be low.
These thoughts lead us to the conclusion - that even if we do live in the Econ 101 world in which lower prices of youth workers raise demand, it is improbable that abolishing the NMW alone would create anything close to full employment among the young.
There are many other things wrong with the youth labour market: lack of demand, lack of skills, poor matching between skills and vacancies, whatever. The NMW is a lesser problem. Blaming it for youth unemployment is an example of what I've called the 'small truth, big error' rhetorical trick.
The notion that scrapping minimum wages is sufficient for full employment of young people is fancifully utopian.