To figure out why this happens, you need to think about what boiling is, and how it works.
As you would know, the water in the pot boils because its temperature was raised above the boiling point by the flame. This required a net transfer of heat from the flame, through the pot, to the water in the pot. Why did the heat flow in this direction? Because the flame is hotter than the water in the pot, even when the water starts boiling ($T_{flame} > T_{boil}$)
Now, think about the water in the bottle. The only way for it to get heat is through the water in the pot. As long as the temperature of the water in the pot, $T_{pot}$, is less than $T_{boil}$, it is still liquid, and it transfers some heat to the water in the bottle. The water in the pot boils off at $T_{boil}$, and can no longer transfer heat as efficiently to the water in the bottle.
This effectively means that the water in the bottle is restricted to a maximum temperature of slightly less than $T_{boil}$, and that is why it never boils.
Another way to think of this is, there must be a temperature difference for a heat transfer to take place. Since the maximum possible temperature of the pot water is $T_{boil}$, the temperature of the bottle water can never exceed this.
EDIT: Another factor to consider is the low conductivity of glass, which means a high temperature difference is required to let a small heat flux through.
Best Answer
Assuming the first was also tap water at the same temp and the pot was room temperature, then all that can be said given your question is "less than 20 minutes". It depends on the thermal capacity of the pot.
Put another way, the first time you boil the water you have to do two things:
The second time all you have to do is:
What really happens when you put the tap water in the hot pot for the second time is that thermal energy from the pot flows into the cooler water and warms it up. This lowers the temperature of the pot and raises the temp of the water until they are roughly equal temperatures (thermal equilibrium). After this balancing the whole system starts out warmer than the first system did and less energy must be put into the system to heat it to the same boiling point. If the first one took 20 minutes then the second one will take less time. The actual amount of time saved depends on how much heat energy was stored in the pot and that depends on the size of the pot, what it's made out of, etc.