The situation is as in the picture. I've made the resistance small compared to that of the tunnel junction so the thermal noise of the tunnel junction can be ignored, since the thermal noise in the divider is given by the parallel combination of resistance.
I don't understand how these noise sources add up. They are both Gaussian, centered at zero and if they are not oversampled after being filtered at 1 MHz, they appear as white.
Wikipedia has an article on this: https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables but when I look at a noise waveform, being random as it is, it seems that there may be some cancellation between the two sources. In terms of amplitude, it looks like they can't add up.
I have an idea that two incoherent noise sources of the same power add up to 3 dB and two Gaussian sources produce another Gaussian distribution, but I might be wrong.
What would be the RMS value of the combined noise at Vo? Would it have a Gaussian distribution if I sampled it at 10 MS/s after passing it through a brickwall low pass filter at 1 MHz? (the noise is not white anymore)
Edit: so the combined noise is 4 mV RMS, or less than 26.4 mV peak-peak for 99.9 % of the time.
Edit 2: There are some assumptions in my calculation:
shot noise is Gaussian, as an approximation for the Poisson discrete distribution that best characterizes it. More exact calculations need to consider the slight skew of the Poisson distribution
the tunnel junction saturates, so it behaves as a current source at the bias point. Most real tunnel junctions don't saturate, but the approximated noise is essentially the same.