Let's start with a Fermi estimate:
The atmosphere has a mass of about 5.15×1018 kg, 20% of that is oxygen. A rocket launch uses on the order of 106 kg of oxygen. To use up all the oxygen (and assuming no oxygen is replenished by plants) requires 1012 rocket launches.
Of course, rockets are only a tiny part of all oxygen consumption. We burn 4×1012 kg of oil every year, and 7×1012 kg of coal. This requires on the order of 1013 kg of oxygen.
Oil/gas/coal/wood burning has a measurable effect on our atmosphere. CO2 concentration was stable at 280 ppm until the year 1800, then it began to rise to the 400 ppm we have today. The total plant mass on Earth is no longer able to convert all the CO2 we create back to oxygen. This also means we have to consider the total oxygen use by the rocket, not just the fraction that is used outside the atmosphere.
Assuming all oxygen is converted to CO2, the oxygen content has changed by the same amount, from 209,580 ppm in 1800 to 209,460 ppm now.
For the rocket industry to have a similar effect on our atmosphere would require 106 rocket launches per year.
The history of our atmosphere gives some hints of what might happen when the oxygen content drops. The O2 fraction was held at 10% until the soil and oceans were saturated. I suspect this means oxygen trapped in the soil and oceans will start to be released once the O2 fraction drops below 10%.