"David A Stocks" writes:

I don't recall how far south of the station the LO track joins the
route from London Bridge. In perfect world there will be signals at
the south ends of both platforms, and only one of them can be cleared
from red to allow a train to proceed. Your problem will be that when
you arrive both trains have a red signal and as soon as one signal
clears that train will take off.

And there would be an indicator, like those at some 'underground'
stations indicating which train will be departing first.