Why doesn't the current stop in neon lamps when all the atoms have been ionized?
If current is the flow of charge why doesn't it stop once all the atoms have been ionized?


The short answer is that it is impossible to ionize all the gas at once. It just requires huge amount of energy to ionize the whole atoms at once. Just for the sake of discussion, even at 10000 °C, still a very small fraction of atoms is ionized! This temperature is hotter than the surface of the Sun! A neon discharge tube is not even close to those temperatures or let us say energies.

The red glow of neon tubes you see is mainly due to electronically excited neon atoms. The pressure of the gas is pretty low inside the tube. Once you start applying a voltage, electrons emit from the cathode and ionize the gas forming a a number neon ions. These ions and free electrons are responsible for carrying the current. Since you are constantly supplying current, it continues to flow through the discharge tube and the process of ionization and recombination continues. Remember that ionization is an equilibrium process in the gas phase as well. An additional process can also occur, that the electron is promoted to a higher energy orbital instead of leaving the atom. Once it jumps back to its original orbit, it emits its characteristic wavelength of light. 

I think the recombination process stated above should yield a continuous spectrum so the red characteristic glow of neon tubes is not from recombination process.