First of all, there is a difference between a monk, or a nun, and a person in consecrated life more generally: monks and nuns belong to cloistered orders and don't generally go "out into the world", whereas sisters and brothers belong to orders which do work "in the world". In either case, though, there is generally a process by which a person joins:
- Inquirer (previously this stage was known as postulancy): In this stage a person is trying to discern, led by the Holy Spirit, whether or not religious life in one specific religious institute is in fact their calling. If they and the local superior of the order agree that this is in fact the case, the inquirer proceeds to the status of
- Novice: The novice is spending a specific amount of time (between 12 and 24 months, according to Canon 648 of the Code of Canon Law) living out the Rule of the religious institute—trying out whether they have what it takes to live as a fully professed member. If, again, they and their superior agree, they next become a
- Fully professed religious: A monk, nun, brother, or sister (or in certain cases, a priest).
It's at the stage of novice that the Code of Canon Law intervenes to state who may and who may not become a novice. In particular, Canon 643 establishes, among other prohibitions, that
The following are admitted to the novitiate invalidly: ... a spouse, while the marriage continues to exist.
Now, you ask about the Pauline privilege. The point of the Pauline privilege is that it dissolves the marriage:
A marriage entered into by two non-baptized persons is dissolved by means of the Pauline privilege in favor of the faith of the party who has received baptism by the very fact that a new marriage is contracted by the same party, provided that the non-baptized party departs.
(Code of Canon Law, Canon 1143, Section 1)
Thus, since the marriage is dissolved, either party would be free to enter a religious institute (assuming there were no other impediments).
Finally, with respect to the case of Nicholas of Flüe which you mention, there are two things to consider:
- This case occurred during the 15th century, well before the current Code of Canon Law took effect.
- Nicholas became a hermit, not a member of a religious institute. He obviously separated from his wife and apparently stopped having sex with her; it's not clear how a modern ordinary would respond to a person attempting to do the same thing today.