First off, in order to answer this question, one first needs to decide what it MEANS to be moderately proficient on the piano, and then someone could determine a time frame for this marker.
To be proficient at anything is largely subjective depending on the activity. If I am proficient at reading, I can read at a particular skill level high enough to comprehend a variety of books and other writings. If I am proficient at playing the piano, I may not be adept yet at playing all types of music at varying levels, because playing an instrument is dependent on a certain mastery of physical and technical abilities. These physical and technical abilities just aren’t really needed when it comes to reading, so again, this is a subjective question.
But let’s establish what it means to be moderately proficient at the piano, and then we can determine how long it will take. To be moderately proficient at piano, a student will need to be able to master several things. First off, they need to know many of the musical symbols used in standard notation such as the rhythmic durations (quarter, half, whole, dotted hald and even eighth notes), dynamic markings such as Piano and Forte which tell the player to play soft or loud.
There are other basic aspects of notation that are important to learn, such as bar lines, the lines and spaces that make up the musical staff, treble and bass clef symbols (and how they translate to the range of the notes on the keyboard), the musical alphabet (A, B, C, D, E, F, G) and even getting into understanding how to name the black notes which are sharps and flats. Finally a student would definitely need to understand the concepts of rhythm and tempo as well as be able to read music and play with both hands. The level of complexity of the music is less important as is the ability to be able to start a piece from the beginning and be able to read through the music on one’s own, with limited aid from their teacher.
With this level of knowledge, a student may be able to sightread some passages from different types of repertoire without the aid of their teacher, or without writing in any note names. Once the student is playing at this level regularly without having to resort to writing in notes, then it’s possible to consider the student as being proficient. This doesn’t mean that they can play any type of repertoire at any difficulty level yet, but it does mean that they have a solid foundation to continue their studies. At this point, they will be about 8 months to a year into lessons.