Let H be graph obtained from $G$ by replacing every edge by path of length $k$. Find number of spanning trees of graph $t(H)$ in terms of number of spanning trees of $t(G)$.
I noticed if $G$ is tree then $H$ is also tree and $t(H)=t(G)$. I tried to find that number based on number of cycles in graph but that didn't help.
Update $1$: Recursive formula crossed my mind $t(G)=t(G\cdot e)+t(G-e)$
Update $2$: I know answer if $G$ cycle because then $H$ will be also cycle and then $t(H)$ will just be length of cycle
Update $3$: If $G$ is tree then $H$ is also tree so in that case $t(G)=t(H)$
This is not an answer to the question. I make Computer simulations: https://github.com/miaosiSari/spanningtree and simulate three cases: (1) $K_n$(2) $K_{m,n}$(3) random graph. Feel free to report a bug!
Currently I am unable to deduce a closed-form solution. It is hard to use the formula $t(G) = t(G \cdot e) + t(G - e)$ here because you have to shrink "too many" edges. You may consider special cases first, e.g., complete or $n$-parite graphs. Or, you may try using computer programs to simulate more situations.