Why did the French Wars of Religion not begin sooner than 1562?

There are several factors which contributed to this.
Protestantism in France did not really take off until the 1540s. The Reformation only began in 1517, with Martin Luther's 95 Theses. The number of French Lutherans increased during the 1520s: at first they were tolerated by Francis I, who was sympathetic to religious heterodoxy, but after the Affair of the Placards in 1534 Francis issued the Edict of Fontainebleau which codified persecution of Protestants. When he died in 1547, his successor Henry II continued these persecutions, which increased in scale. Thus, it took some time for Protestantism to become entrenched in French society.
Lutheranism never gained a strong foothold in France; the Lutheran presence was scattered and disorganised. It was not until John Calvin organised and formalised the structure of his sect of Protestantism - Calvinism - in 1541 and deliberately targeted France for conversion that significant progress was made. Calvin had religious texts and pamphlets, written in French, smuggled across the border from his safe haven in Geneva into the hands of the French nobility. Therefore, it was not until the 1540s that a clear strategy for strengthening Protestantism in France was implemented.
Before the death of Henry II in 1559, France was ruled by strong monarchs who commanded the support of the nobility, estates, and clergy; after his death, it was ruled successively by his three sons, who were in various ways seen as weak and unfit to govern, and thus they held a great deal less power and were less able to control religious tensions.