Twisted convolution - simplifying multiple integral via substitution
Since these are two different integral equations, during substitution you cannot use the same variables for both the integrals.
On substituting $r_1(t-\tau)$ to $r(t)$ we get,
$$r(t) = \int \int h_2(\tau,\nu)e^{j 2\pi\nu(t-\tau)} \int \int h_1(\tau_1,\nu_1)e^{j 2\pi\nu(t-\tau-\tau_1)}s(t-\tau-\tau_1)d\tau_1 d\nu_1 d\tau d\nu$$
Now substitute $\tau = \tau', \tau_1 = \tau - \tau', \nu = \nu', \nu_1 = \nu - \nu'$