How the bubble size in a champagne bottle evolve?

When the bottle is closed, part of the gas is dissolved in the liquid. The portion that is not dissolved is responsible for the pressure. Any additional gas leaving the liquid increases the pressure, what prevents the continuation of the process.

When it is opened, the pressure is atmospheric, and the fraction of stable dissolved gas is certainly much smaller that the actual content.

But for the molecules of gas form a gas phase (a bubble), it is necessary also to form an interface liquid-gas (the bubble spherical surface).

Where only very few molecules gather, the free energy to escape to form a gas is not enough to form the surface (that requires some energy), and they return to the dissolved status.

Where their number is big enough, the critical radius is reached, and a stable bubble is formed. Once that happens, the dissolved gas in the neighborhood can migrate to the existing bubble, that grows as a consequence.

It is a process of nucleation and growth, common to many changes of phases.