MacMillan and I agree in disagreeing that not to mention and let alone are synonymous:
not to mention: used for adding a comment that emphasizes the
main idea of what you have already said: The weather here is
gorgeous, not to mention the wonderful food.
let alone: used for saying that something is even less likely
to happen than another unlikely thing: I hardly have time to think
these days, let alone relax.
still/much/even less: used after a negative statement in
order to emphasize that it applies even more to what you say next: I
am no one's spokesman, much less his.
You could make the case that let alone and much less are synonymous, since you could substitute one for the other:
I am no one's spokesman, let alone his.
I hardly have time to think these days, much less relax.
However, substituting not to mention and let alone for each other would result in sentences that sound contradictory:
The weather here is gorgeous, let alone the wonderful food.
I hardly have time to think these days, not to mention relax.
I find the distinction that MacMillan makes between not to mention and the supposedly synonymous let alone and still/much/even less useful:
- The phrases let alone and still/much/even less reinforce a negative or unlikely statement that precedes them.
- The still/much/even less constructs reinforce the negativity of the preceding phrase by subtraction -- Negative statement, still/much/even less.
- In a similar manner, let alone reinforces the unlikeliness of the preceding phrase with an even more unlikely scenario -- I couldn't get a date with a reanimated corpse, let alone the prom queen.
Some grammarians insist that not to mention and let alone are synonymous; while you can find some cases where substituting one for the other might make sense, I generally find their meaning and usage distinct from each other.