I went through traditional on-campus MBA recruting where all interviews (7 firms, those three plus 2 other strategy consulting, one economic consulting, and one industry) were within 1.5 weeks and classes were cancelled, so we generally had several back-to-back days of interviews. I'll give you my perspective based on that.

I'd try spread them out only slightly, such as try to do all three first rounds over 2 weeks.

Take it with a grain of salt, but if I could plan out my ideal schedule it would be something like this:

Week 1: Tuesday - Mck round 1. Then a final round if it happens likely later that week. This week is entirely Mck focused since Mck is different than Bain/BCG interviews, but ideally not back to back days (sometimes you don't have a ton of control over that)

Week 2: Monday - Bain or BCG round 1, final round on Friday. On Wednesday - other Bain/BCG round 1. Since there is some flexibility in final rounds esp if they fly you out to their location, try not to do an interview on one day and have to fly that same day for a final round in another city the next day - that's pretty tiring.

Week 3: Final round for Bain/BCG whenever possible.

Some thoughts/rationale/answers to your questions:

Don't put them all in the same week because then you run into potential overlaps on second/final rounds, stressful flights (if the firm does 2nd rounds in the specific city you are applying to, etc), and the dreaded multiple firm interviews in the same day - that is completely exhausting.

However, I'd also advise against spacing them out much more than that. You should get into your top form before the first one - you shouldn't have any major to work on in between interviews and therefore getting into the mental state of mind and maintaining it for over a month would be very difficult. You can only maintain that mental intensity of focus for so many weeks before you just stop caring. In between interviews, you shouldn't be doing intense case prep - maybe just reviewing concepts, ideas, light practice drills, review cases you've already done, etc.

The schedule above gives you enough time if you hypothetically get a Mck offer to wait until seeing what happens with BCG. And if you get multiple offers, they will all bend over backwards to accommodate your timing preferences.

Now, everyone is slightly different in how much they can handle the trade-offs above, but the impact of both extremes are very real (too compressed being way too stressful and mentally exhausting, and too spread out making it difficult to maintain your focus for so long).

Best of luck!