Assume that $n\geq p$. Let $\phi:X\in M_{n,p}\rightarrow ||X||_*$; then $\phi$ is $C^{\infty}$ in a neighborhood of any $X\in M_{n,p}$ that has full rank $p$.

Let $f:A\in S_p^{>0}\rightarrow tr(\sqrt{A})$; then $Df_A:L\in S_p\rightarrow 1/2tr(LA^{-1/2})$.

According to my post or that of greg in (1)

Derivative of the nuclear norm with respect to its argument

The derivative of $\phi$ is $D\phi_X:H\in M_{n,p}\rightarrow tr(X(X^TX)^{-1/2}H^T)$.

Unfortunately, there is no such closed form for $D^2\phi_X(H,K)$, because, in general, $X^TX$ and $(X^TX)'$ do not commute (that is useless for the calculation of the first derivative).

On the other hand, that is above does not work when $X$ has not full rank. Since the norms are all equivalent, I do not understand why researchers persist in using the nuclear norm instead of the standard Frobenius norm. Something escapes my mind but the specialists surely have good reasons.

About alternative methods, Peder gave here an interesting answer concerning the sub-differential. Moreover, Michael Grant in (1) again, speaks about several methods that seem to give a good answer to the OP's question; the last but not the least, Michael gives a link to his software TFOCS

http://cvxr.com/tfocs/

Finally, specialists get tired of answering questions often without really being listened to; indeed, people prefer to reinvent the wheel...