Are Catholics allowed to leave marriage in order to become a monk or nun?

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:

1. This case occurred during the 15th century, well before the current Code of Canon Law took effect.
2. 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.

From Roman Cholij's Priestly celibacy in patristics and in the history of the Church:

Although perhaps strange to our own modern ways of thinking, absolute marital continence was far from unknown or unesteemed in patristic times. Tertullian, himself a married man, informs us in his Catholic period, of lay people who practise continence within marriage «pro cupiditate regni coelestis».¹¹ So do Jerome and Augustine in the following century.¹² The rapid growth of monasticism and an attraction to the ascetic life led many couples to renounce their intimacy and to enter a monastery¹³ or to live in continence within more domestic settings. Church authorities had to intervene decisively when the enthusiasm for continence was deemed excessive and tainted with heretical motives, but at the same time praising those who lived the life of continence for the right motives.¹⁴ Four centuries later the Second Nicene Council (787) would still endorse the possibility of monastic vocations for the married.¹⁵

The relevant part from the Second Nicene Council (787), Canon 20, reads:

If there are persons who wish to renounce the world and follow the monastic life along with their relatives, the men should go off to a male monastery and their wives enter a female monastery, for God is surely pleased with this.

It would seem 1983 Can. 643 §1.2/ ("The following are admitted to the novitiate invalidly:…a spouse, while the marriage continues to exist;") or the corresponding 1917 Can. 542 1.° [d] abrogate the Second Nicene Council (787), Canon 20; but, as the commentary on the 1983 Code says,

if a couple mutually agrees to renounce marital rights and relationships in order to seek admission to religious institutes, a dispensation from the Holy See must be sought.

A famous recent example of a married couple separating—with the pope's permission, the husband becoming a priest, the wife becoming a nun, and their biological children going to boarding school—is that of Mother Cornelia Connelly (1809-1879), foundress of the Society of the Holy Child Jesus, a sort of modern-day St. Rita of Cascia. See The Life of Cornelia Connelly (1922) or this video documentary on her.

For a history on this, see Spiritual Marriage: Sexual Abstinence in Medieval Wedlock; its appendix gives a listing of many spiritual marriages among the canonized saints of the Church:

Also, pp. 2-3 of Validity of Virginal Marriage by John Cuthbert Ford, S.J., contains a list of married virgin saints.