How long does it take for the tour boat to get from the Tower of London to Westminster Abby or at least close to it?
Impossible to answer without knowing what tour you are talking about.
It is actually a tour or is it just the sightseeing boat from the Tower to the Westminster Pier area? They do have a narration on the boat in several languages. It takes about 35 minutes to travel between the 2.
The River Thames water transport
Agree. About 25-30 minutes.
In regards to your desire to be spontaneous it would beneficial to check websites of locations regarding opening and closing hours. Avoid disappointment.
I'm not certain if you are referring to the Uber boats, so if not please disregard. You will want to keep in mind the time to actually board a boat with available room. When we were in London in April Tower Pier was incredibly overloaded midday, such that we had to wait for 4 or 5 boats to come before we got yo the front of the line to board.
I greatly underestimated that waiting time and it added some extra stress when you're already hot and tired.