Hi there,
Firstly, HUGE CONGRATS on getting the offer!! I hope you took time to be proud of yourself and celebrate!
Flexibility of start date is mostly dependent on the office (country/city). In general MBB is very flexible on start date - when they find someone they want to hire, they tend to be patient. However, some offices have fewer "start dates" than others, and some may really need new consultants ASAP to meet a recent surge in demand.
My recommendation:
There is 100% no harm in asking. I would just contact the recruiter / HR person you've been in contact with and tell them that you'd like to start a bit later. Most likely, it will be doable, and if not, it still won't offend anyone that you asked
I hope this helps!
Thor