T2 to MBB - When to Lateral

I had someone else on this forum ask something similar so I'll add my two cents with your specific case. First, prefacing with the fact that what I'm about to say is from second-hand experiences from 20-30 people I've spoken to from friends to colleagues. 

Lateral Consulting Position Route to a Different Company

• Below 1.5-2 years: Many consulting companies consider anyone below 1.5-2 years of full-time experience to be a recent graduate. If you apply through this route, your market research experience may be discounted (you start at the new-grad entry level). If this is the route you plan to take, then you can start networking or recruiting immediately and target the traditional new grad timeline (Sept-Oct recruiting season). If you target firms within the sectors you cover (Huron/LEK may be good options), then you would have a strong shot. 
• At or Above 1.5-2 years: After the 1.5-2 year mark, many firms consider you as an experienced hire and put you into a separate pipeline. These positions are generally seeking 1-3 years of experience, hire on a needs-basis typically throughout the year. If you're considering applying through this route, it's best to reach out to the companies to get an understanding of the hiring needs, the type of projects and skillsets since the hiring is more opaque than the new grad sequence. 
• Your Situation: If you plan to lateral below this point, understand that firms have different standards/considerations. For example, BCG considers anyone outside of ungrad/grad to be experienced, so you would be considered against applicants at 2-4 years out. MBB altogether aims to hire the top 1%, so expecting to get an interview below 1.5 years, even with a referral, will be difficult. 

Lateral at Your Current Company

• The Challenge: The company trained you for the role you're currently in so you can gain a specialization and be efficient in the work. If you want to switch to consulting, you have to make a really strong case for it. It depends on what the other team within your company needs (do they have a need for a new consultant), what skills they're looking for (technical skills like coding or database languages, slide or deck construction, modelling capabilities) and whether your current team is in a position to lose you (are there other colleagues that could absorb the work). The trend I've seen at the Big4 firms when colleagues try to rotate from Audit to Consulting is that they're typically met with resistance. A few have even mentioned it'd be easier to apply to their competitor for consulting than to switch departments internally. 
• The Outcome: Of course, it does happen, and I'll explain how. Often, these firms have a "coach", "buddy" or "HR Lead/Resource Coordinator" you can speak to (as well as a supervisor/manager) about the feasibility of moving to a different team. I would argue this process can take anywhere from a few months to a year depending on how essential or short staffed your current team is and whether they're willing to move and retrain you for the other department. Before kicking off the formal process with HR, I would advise you network internally to confirm that they do in fact need someone in the consulting team and ask for any skills you may want to develop to prove interest & competence.
• Your Situation: Perhaps look to internally network with more strategy oriented work. Most consulting shops are fluid with their assignments and clients. This would let you build a more relevant skillset (if you're explicitly targeting strategy consulting-type work) and give you insights into the work.  

Hope this helps.