In cases such as this, the proper thing would be to let the person who scheduled the date first keep it, and the other person should reschedule. To not do is very rude and could be a power play or dis of the new bride.
Of course, there are always exceptions. Maybe one or the other has a schedule that is not flexible, or has already placed deposits on wedding venues, etc. and the chosen date cannot be changed
In either case, there should be a calm, non-judgemental discussion between your groom and his mother.