This question is impossible to answer with any degree of accuracy. Largely it will depend on the level of your desire and commitment. By establishing a daily practice regime and following guidance carefully, you should achieve noticeable progress on a weekly basis. If you are a complete beginner, this means you should expect to be playing recognisable tunes within a few weeks, and soon be adding harmony. As you become more fluent with reading notation and coordinating fingers, your teacher will add new notes to your range and introduce new musical devices.
Most tutor books work on the principle of introducing something new at each lesson. In practice, however, many students find that they will need to stick at a particular level for a little while to properly assimilate and develop what they have learnt so far. This is perfectly normal, and your teacher will be able to provide supplementary material to fill these ‘gaps’.
Grade examinations are designed specifically for this purpose too, and mark a recognised standard of achievement at each level. Typically, if you are a piano student beginning from scratch at the age of 7-9, you could expect to reach Grade 1 (ABRSM) standard within two years. Older children may achieve this quicker, and adults vary a great deal, depending on a number of factors, such as whether they have studied an instrument before, and how much time they are prepared to commit to learning.