> H0(H1(m)) has the security of just H0. Hashes are not made to protect the content of m, but instead made to test the integrity of m. As such, a flaw in H0 will break the security guarantee, no matter how secure H1 is.
But this isn't true for all flaws. For example, even with the collision attacks against SHA-1, I don't think they're even remotely close to enabling a collision for SHA-1(some_other_hash(M)).
Similarly, HMAC-SHA-1 is still considered secure, as it's effectively SHA-1-X(SHA-1-Y(M)), where SHA-1-X and SHA-1-Y are just SHA-1 with different starting states.
So there's some value to be found in nesting hashes.