Random signal power spectrum
You have to simplify your expression in order to obtain a Dirac Comb (a.k.a. impulse train).

$$R_X(t+\tau,t)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty} \delta(t+\tau-n \tau)\delta(t-m\tau) = \sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty} \delta(t-(n+1)\tau)\delta(t-m\tau)$$

Both $\sum_{m=-\infty}^{\infty} \delta(t-m\tau)$ and $\sum_{n=-\infty}^{\infty} \delta(t-(n+1)\tau)$ are actually the same signal: a Dirac Comb with $\tau$ period. The difference would be the that the ''n'' signal would be time shifted by n+1, while the ''m'' signal is shifted by m. As they are defined in $(-\infty,\infty)$, they are defined in exactly the same instants.

Applying delta's well known property:
$$\delta(t-t_0)x(t)=x(t_0)$$

we can obtain that, as we intuitively know because both signals are the same, 
$$R_X(t+\tau,t)=\sum_{m=-\infty}^{\infty} \delta(t-m\tau)$$

From here, we just have to obtain the Fourier Transform of the Dirac Comb, which can be easily found, e.g. in Wikipedia, obtaining that all the coefficients are $\frac{1}{\tau}$.