It looks like we have similar backgrounds (though we work in different industries) and similar goals, but I'm further along the path than you are. I've been actively preparing for the past few months and am happy to share my experiences with you; if you'd like to discuss in more detail, feel free to message me directly.
As someone who's going through the process currently, I want to emphasize something that others have said: your biggest challenege will be securing an interview. Because you are coming from outside the traditional recruiting pipeline (i.e. universities), you will need to a referral from a current consultant in order to get your application reviewed. Ideally you can leverage your existing network (e.g. old labmates, university alumni, etc), but if that fails you should try a combination of LinkedIn and attending recruiting events.
Regardless of the channel used, once you get in touch with an MBB consultant and establish a good rapport with them they should be able to give you targeted advice on your cover letter, resume, preparation strategy, etc. Your cover letter becomes extremely important at this stage, because you will use it to justify your career change. Recruiters will be asking themselves "Why does this candidate want to enter consulting now?" and a good cover letter will address that question.
When putting together your application materials and thinking about what to emphasize, think about it from the recruiter's point of view: if they see a PhD candidate with 4+ years of experience, what will they think about your strengths and weaknesses? They will probably assume you have strong analytical skills, but may lack leadership and interpersonal skills. They may also assume that you don't know much about consulting, nor why it fits with your career goals. Given this, it becomes even more important that you use your application to address those perceived gaps (e.g. leadership, fit, managing conflict, etc.).
Finally, as others have said...Start practicing cases! It's never too early to start, and take it from me - practicing with a full-time job is very difficult. The more time you give yourself to do it, the better.
Thanks Andrea for your inputs.