I would not recommend you wait the invitation to start preparing on cases.
Many candidates get max 2-3 weeks notice before the first round. Also, many candidates need a minimum of 30 cases to be in good shape for a consulting interview. Plus, there is the time to read the theory, drill on math and graphs, prepare the fit, etc.
All in all, many candidates need 100h+ to be ready for the interview. If you try to push that in 20 days, it would be 5 hours per day. It is feasible, but not ideal.
As a consequence, if you are 100% committed to join a consulting firm, I would recommend you start in advance. The best candidate I met did start preparing at least 2-3 months before the interview.
Even in the bad scenario where they don’t invite you for interviews (which is unlikely to happen if you follow the right steps, in particular in terms of referrals), you will have learned some pretty valuable skills on problem-solving.
Finally, as a mindset, the act of investing in advance time in your preparation will strengthen your commitment and increase chances and opportunities to succeed. It is easier to succeed when you are all-in.
In terms of what to do for the preparation, you can find some tips at the link below:
Please feel free to PM me for more information.