1. Historically, the labor pains would be when the Apostles were setting up the Church, traveling, working miracles, etc. And also the post-Apostolic age, wherein the Church was directly fighting paganism.
2. Spiritually, the labor pains would be whenever a convert hears the truth and thinks about converting. The devil waits until the person is "born again" and then (many times) God allows the new Catholic to experience trials and tribulations, to increase/strengthen their Faith.
3. Futuristically, the labor pains would be during the period of time between popes (or good popes), when the Church/Our Lady is waiting to "give birth" to a new pope. In our times, obviously, this could refer to the difficulty that the Church has had in producing an orthodox leader for the last 100 years. To continue the labor analogy, a holy pope is "overdue" and the Church/Our Lady is in pain to provide the world with a holy leader.
Further, it could also refer to some great, internal conflict in the Church, a cινιℓ ωαr (i.e. post Francis' reign) where the TLM-loving clerics finally step up and become in conflict with the Modernists, leading to a prolonged conclave, wherein Our Lady finally prevails and a holy pope is elected. Prophecy speaks of a series of short-lived holy popes before we get THE Angelic Pope who rules with the Holy Monarch in the restoration.