The resulting graph is the linegraph of the subdivision graph of $G$. This survey paper of Bojan Mohar tells how to obtain the Laplacian spectrum of the linegraph of a semiregular graph and the subdivision graph of a regular graph.
Let's generalize. Let $G$ be a regular graph of $n$ vertices, degree $d$, and therefore $m=nd/2$ edges. Let $\mu(G,x)$ denote the characteristic polynomial of the Laplacian matrix, and let $\kappa(G)$ be the number of spanning trees.
The blowup $B(G)$ of $G$, formed by replacing each vertex by a $d$-clique, is the linegraph of the subdivision graph of $G$. Using Theorems 3.8 and 3.9 in the survey paper of Mohar, we find $$ \mu(B(G),x) = (-1)^n (x-d)^{m-n} (x-d-2)^{m-n} \mu(G,x(d+2-x)). $$
We know that $\kappa(G) = n^{-1} (-1)^{n-1} \mu'(G,0)$. Differentiating and using $\mu(G,0)=0$, we find $$ \kappa(B(G)) = d^{m-n-1} (d+2)^{m-n+1} \kappa(G). $$
For the $k$-fold blowup, we have $$ \kappa(B^k(G)) = d^{d_k(m-n)-1d^{d_k(m-n)-k} (d+2)^{d_k(m-n)+1} d+2)^{d_k(m-n)+k} \kappa(G), $$ where $d_k=1+d+\cdots+d^{k-1}$.
For $d=3$, I believe this gives $$ \kappa(B^k(G)) = (5/3)^k 15^{(3^k-1)n/4} \kappa(G). $$
