All else being equal, the odds of being called back for an interview are greater:
(1) the more senior your contact is: A partner will have greater pull than a 1st year associate
(2) the better your contact knows you: Best is you have worked together in the past, least is you just said hi once 5 years ago
(3) the closer your conctact is to the actual decision maker, typically a PL/Principal. Obviously if they are close friends, the recommendation will have more weight. We are all humans after all, even consultants can be driven by emotions
Assuming you do apply through that contact, and he convinces the recruiting team to bring you in, you will still have to submit a normal application of course - but by doing so, you ensure your resume is looked at with the proper lens rather than summarily dismissed along with dozens of other candidate lambda applications.
As for the timeline... it will vary. Most often you should hear within a few days to couple of weeks, but it can be much longer on occasion: A Senior Principal recomended me on the strength on our work together on a recent project (I was the client, and worked daily w/ BCG for about 4 months), yet I didn't hear back form BCG for a full month, and my1st round interview was scheduled another month out. The wait was torture, but ultimately worth it :)
Good luck w/ the application. Oh - and don't wait to start preparing the case interviews. The last thing you want to do is get an interview in 4 days and not be ready!