How to calculate the noise power for a non-stationary noise?
If your signal is just $n(t)$ distributed as $N(\mu, \sigma^2)$, then the expected signal power is then:
$$
p_{\rm ensemble} = E\left [ \left| n(t) \right|^2 \right] = \mu^2 + \sigma^2
$$

For the time-varying case, I'd suggest that the answer is:
$$
p_{\rm time\ varying} = \mu^2(t) + \sigma^2(t)
$$