Fastest way to get an associate position at McK is to get a BA position at McK & do a 1 year MBA (or get an early promotion, an extreme rarity, even for high performance, often more of a political promotion).
MBA is a perfectly fine track for getting an associate position at McK, not every MBA has work experience. Some get straight into an MBA program out of undergrad, which means you see MBA's entering at the Associate level around 22-24 if they're quick movers, but usually around 26-30. Normally, if you have a McKinsey associate at 25, he/she's a BA-hire who got the MBA paid for by McK
Law school is not the fastest track to an Associate position. However, its a perfectly fine track. HYS are all respected law schools, I tend to interact with more H&S than Y but thats besides the point. Just graduate cum laude and you should be able to pull an interview.
Many of the JD/PhD's I see at McKinsey work in fields outside their graduate degrees. They all have distinguished degrees and did good work, but they generally pick a specialty / field at McK which is unrelated to their area of study. Occasionally you will see some overlap between their assignments and education, but even then the overlap is generally pretty small, not enough to warrant an additional 2-3 years. It makes much more sense to get an MBA, time-wise, career-wise, debt-wise.
I would tell you that if you want to be a lawyer, or like law exit opptys in case you tire of mgmt consulting, then go the JD route and see if it works. However, if you are studying to be a lawyer to get an Associateship at MBB then stop what you're doing right now. There's nothing special that JD adds to your resume or exit oppts as a consultant unless you want to focus purely on legal practices.