It is a partial answer for your question:

For $P_n$, the form of normalized laplacian matrix is three diagonal and with some calculations, we can show that all its normalized laplacian eigenvalues are simple. For example, the normalized laplacian spectrum of $P_n$ for $n=2,3,4,5$ are:

$P_2:$ $[0,2]$,

$P_3: [0,1,2]$,

$P_4: [0,2,\frac{1}{2},\frac{3}{2}]$,

$P_5: [0,1,2,1-\frac{\sqrt2}{2},1+\frac{\sqrt2}{2}]$.

In the paper with name:

"Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications",

there are some useful results about fractal and Cayley tree and their normalized laplacian spectrum and their degeneracy. Specially, in section $III$ part $B$, you can find some useful calculation. For example, the fractal $F_1$ for $m=0$ has simple normalized spectrum and is an other example for your question. I wrote a Maple program that can calculate the requested spectrum. If it is useful, I can send it for you.