Switching input BNC signal to two output BNC signals using BJTs?
It will not work (reliably) like this.

The transistors turn on when their base voltage is below their emitter voltage.
But here the emitters are DC coupled to the PMT output.
So for this to work the PMT output needs to have a DC-component at the TTL high level. 

I would consider:


Looking for a relay/switch with BNC connectors
Seeing if CMOS switches like CD4066 will suffice
Redesign