Both bodies of water will heat till 100 C (as well as the vessel). Then, the outside one will start boiling by taking latent heat from the surroundings, Now, we only have heat transfer due to conduction/radiation when there is a temperature difference (or an emf, but that's irrelevant). It's mainly conduction we need to consider, anyways. Convection won't get heat across a solid. Now, there is no temperature difference across the vessel, thus no net heat transfer. If we placed cooler water inside, there would be enough heat transfer to bring it to 100 C, and then it would stop (we can't boil pure water below 100 C at 1 atm)
So the sequence of events is as follows:
- Outside water starts heating
- Inside water and vessel start heating; with a small time lag
- Everything reaches 100 C (with a small time lag)
- Outside water starts boiling
- Outside water is completely boiled
- Outside water(now steam) heats up an infinitesimal amount
- Outside water continues to heat up, but also transfers heat to inner vessel. The vessel heats up and the inner water starts boiling.
- Inner water finishes boiling. Outside steam will have reached a much higher temperature (can't be calculated without knowing the masses)
Of course, since it's steam, it will probably diffuse out, so eventually only a tiny bit of the inside water will boil.
In fact, this is one of the ways one can keep a fixed mass of water at exactly 100 degrees celsius. Useful for making extremely accurate calorimetric calculations.