Yes you should pay twice if using cash or PAYG. There are no "break of
journey" facilities for London's buses nor are there are through single
tickets valid bus to bus. Obviously you aren't really paying twice if
using a Travelcard and if on PAYG you are simply contributing to meeting
the daily cap so perhaps not a great hardship.

How about if the bus terminates earlier then it was advertising when you got
on it?

Do you then still have to pay again for the next bus through to your
destination?

LEWIS