How long will it take to develop natural vibrato?

You need another voice teacher. There are unfortunately a lot of people "teaching" singing that shouldn't be trying to teach pigs how to squeal.

No, you do NOT just SHAKE your voice. And a "natural" vibrato is ALWAYS in your voice. It's not something you put on and off like a sweater. A natural vibrato should be the result of healthy singing. Sometimes the vibrato can be very subtle, sometimes very prominent--however if it's the first thing you notice about a voice--it's too much. Literally, there is a slight fluctuation of adjacent pitches in a voice with a vibrato. HOWEVER, the impression should be that the singer is singing perfectly ON pitch, not veering in and out of tune.

There are certain vocal effects that involve deliberately and rapidly alternating between close pitches on like trills (usually used only in classical singing as in opera). That is not a vibrato either.

If your voice teacher cannot answer your singing questions or explain concepts and techniques related to singing (which I suspect she has no clue), then you are wasting both time and money. You won't find your answers here on Yahoo either. People are just likely to tell you some nonsense about diaphragms.

Your entire voice must be trained as a whole, not in "parts". You don't learn vibrato one day, and next day do "high notes" and the next learn breath control. A good voice teacher can certainly assign songs or exercises to allow you to practice a certain technique, but again--you don't teach someone isolated techniques. You teach them how to sing.

It really depends if your teacher really is teaching you proper technique. Remember, there are a lot of bad teachers out there. What are you learning? Theatre, SLS, classical? Those are three examples, but what method or style does your teacher teach? I find that very typical with SLS teachers.

First of all, it takes YEARS even with learning the foundations of singing and correct breathing techniques, so please do NOT try to rush too much. You NEED to be patient and you NEED to have a GOOD teacher. First things first, and in case your teacher isn't a good one, you NEED to find yourself a better teacher before you cause any laryngeal damage for yourself.

Vibrato is just shaking the voice... you just do it... and with practise it sounds better... and then eventually you just do it without noticing which i hate i don't like it when people can only sing with vibrato