Paillier Private $\mu$ and $\lambda$
What information does $\mu$ leak about $\lambda$?


The safe assumption is: all. It must be assumed that knowledge of $\mu$, together with the public key, allows computing $\lambda$ (which allows decryption and factorization of $n$).

At least, that holds in Paillier's scheme as described in Jonathan Katz and Yehuda Lindell's Introduction to Modern Cryptography (section 13.2.2). In this we have $p$ and $q$ of equal size, $g=n+1$, $\lambda=(p-1)(q-1)$, and $\mu=\lambda^{-1}\bmod n$. It follows that $\lambda=\mu^{-1}\bmod n$, allowing computation of $\lambda$ from $\mu$ and $n$ (using e.g. the extended Euclidean algorithm, which is inexpensive).

While that does not immediately tell how to compute $\lambda$ from $\mu$ and $n$ in Paillier's scheme as in the question, that's enough to show that we can't safely reveal $\mu$.