Your company uses a meeting management system to book meetings throughout the day. Unfortunately, the system does not consolidate overlapping or consecutive meetings, leading to confusion and inefficiency. Your task is to write a function that merges overlapping and consecutive meeting times to streamline the meeting schedule.

Given a list of meeting times represented as tuples of start and end times (start, end) (in 24-hour format), your function should return a new list with the overlapping and consecutive meetings merged. A meeting is considered to overlap if it starts before another meeting ends. Consecutive meetings have no gap between them.