Postponing your start date should not be a problem at all; just explain your situation to your Program Director.
Even if you were to miss a few weeks of your first month block or even a portion of your second block, you will see the material later and you will have those rotations again. Also, the earlier you tell your PD, the better chance of them scheduling you for a rotation during the first month or two that you will have again during your first year (so that you won't miss that experience entirely during your first year). For example, if at your program each 1st year radiology resident does two months of ultrasound, they can schedule you for ultrasound during month one, and that way if you miss it, you still get exposure to it during your second rotation of your first year.
Also, I personally know of a new radiology resident who significantly postponed the start date (over 1 month) because of maternity reasons, without any problems, and I think a death in the family would also be excusable. Your program may require you to apply vacation time to this time off (which I think is the situation for the resident I know).