Yes MBB all have flexibility in starting date (I am not sure about tier 2 and others). It is possible to postpone.
The length would be a case-by-case discussion with the office you are starting with, depending on the new joiner pipeline as well. It doesn't matter much which way you join actually. I postponed my BCG start date by 1 month (MBA). I know someone postponed Bain start date by 6 months (Referral) and another person by 1 year (MBA).
This year with COVID situation, I think it would be even easier if you want to ask to postpone.