It is highly variable. Some have their first interviews within a few days of applying, while others (like me) wait for a couple of months.
I recommend you start your prep as soon as you know you will be applying, knowing full well that the last couple of weeks before the interviews will be very intense regardless. This, for 3 reasons:
1. It takes time to really 'get' the case interview process. Your brain just needs to sleep on it, and craming at the end won't leave you enough time to do that
2. The skills acquired are portable, and will serve you in other interviews as well - so even if you do not end up having an interview with your dream consultancy, at least you will have prepared for future cases. [note: same skills can and should also be used at work btw]
3. Once you have the interviews lined up, you will naturally start feeling a little bit of stress and your prep will become more intense. Best to start that last phase at a higher overall level of preparation, right?