Here are some studies that suggest how much can various types of exercise increase vital capacity of the lungs and in what time.
Effects of swim training on lung volumes and inspiratory muscle conditioning (Journal of Applied Physiology, 1987)
Lung volumes and inspiratory muscle (IM) function tests were measured
in 16 competitive female swimmers (age 19 +/- 1 yr) before and after
12 weeks of swim training. Eight underwent additional IM training; the remaining eight were controls. Vital capacity (VC) increased
0.25 +/- 0.25 liters, functional residual capacity (FRC) increased 0.39 +/- 0.29 liters, and total lung capacity (TLC) increased 0.35 +/- 0.47 in swimmers, irrespective of IM training.
In this study Increasing lung capacity and cardiovascular ability by Mosesahi gymnastics in Gorontalo State University students (Annals of Tropical Medicine and Public Health, 2017)...
...performing "Mosesahi gymnastics" for 40 minutes per day for 3 times a week, increased vital capacity of the lungs in some participants (university students) by as much as 1.2 liters in 3 months (see Table 2).
Lung-packing and stretching increases vital capacity in recreational freedivers (European Respiratory Journal, 2012)
The diver's lung training involved a set of 5 different lung exercises with yoga and lung packing maneuvers 5 times a week for 11 weeks. Mean vital capacity had increased across the training period, from 5.9 to 6.3 L or by 7.5%.
Lung packing = glossopharangeal insufflation or buccal pumping
Swimmers may have large lung volumes because of increased number of alveoli.
The large lungs of elite swimmers: an increased alveolar number? (European Respiratory Journal, 1993)
These findings suggest that swimmers may have achieved greater lung
volumes than either runners or control subjects, not because of
greater inspiratory muscle strength, or differences in height, fat
free mass, alveolar distensibility, age at start of training or
sternal length or chest depth, but by developing physically wider
chests, containing an increased number of alveoli, rather than
alveoli of increased size.