Probability of error of 2 identical BPSK channels
For the system you have drawn (with no possible way to judge which channel is correct) if one is in error then, the probability for system error has doubled. The above is based on the confusing symbol where the two channels are recombined. I initially read it as a multiplier.

If the symbol is an adder then two BPSK signals can add and produce double the signal amplitude whereas the noise signal only goes up by \$\sqrt2\$.

Two noise sources added together can be assumed to be non-coherent hence they add as \$\sqrt{A^2+B^2}\$ hence, if A = B then the two added together increase the RMS by \$\sqrt2\$.

In this case the SNR (signal to noise ratio) has increased by 3 dB but, to convert this to a BER (bit error rate) means understanding what the original BER was: -



As you can see, the graph tells you the BER for a particular SNR but we don't know what the SNR was for a single channel so can't provide a definitive answer but you can use the formula you originally quoted with E/N rising by 3 dB.