What happens to one's sins if the priest omits the words of the Absolution?

In order to answer your question fully well, first we need to understand what gives absolution to a sinner who confesses. The sins of a confessor will not be absolved simply by stating the them. In order to achieve absolution either perfect contrition or partial contrition (also known as attrition) is required. Contrition is sincere and complete remorse for sins one has committed. Attrition is a desire not to sin for a reason other than love of God, mostly fear. Mortal sins are absolved only through the sacrament of penance. But the council of Trent has made it clear that mortal sins are already forgiven by contrition.

Session XIV of the council of Trent has discussed the topic "The Most Holy Sacraments Of Penance And Extreme Unction" and chapter IV of the official document titled "contrition" is given below.

Contrition, which holds the first place among the aforesaid acts of the penitent, is a sorrow of mind and a detestation for sin committed with the purpose of not sinning in the future.[16] This feeling of contrition was at all times necessary for obtaining the forgiveness of sins and thus indeed It prepares one who has fallen after baptism for the remission of sins, if it is united with confidence in the divine mercy and with the desire to perform the other things that are required to receive this sacrament in the proper manner. The holy council declares therefore, that this contrition implies not only an abstention from sin and the resolution and beginning of a new life, but also a hatred of the old,[17] according to the statement: [18] And certainly he who has pondered those lamentations of the saints: [21] and others of this kind, will easily understand that they issued from an overwhelming hatred of their past life and from a profound detestation of sins. The council teaches furthermore, that though it happens sometimes that this contrition is perfect through charity and reconciles man to God before this sacrament is actually received, this reconciliation, nevertheless, is not to be ascribed to the contrition itself without a desire of the sacrament, which desire is included in it. As to imperfect contrition, which is called attrition, since it commonly arises either from the consideration of the heinousness of sin or from the fear of hell and of punishment, the council declares that if it renounces the desire to sin and hopes for pardon, it not only does not make one a hypocrite and a greater sinner, but is even a gift of God and an impulse of the Holy Ghost, not indeed as already dwelling in the penitent, but only moving him, with which assistance the penitent prepares a way for himself unto justice.

And though without the sacrament of penance it cannot per se lead the sinner to justification, it does, however, dispose him to obtain the grace of God in the sacrament of penance. For, struck salutarily by this fear, the Ninivites, moved by the dreadful preaching of Jonas, did penance and obtained mercy from the Lord.[22] Falsely therefore do some accuse Catholic writers, as if they maintain that the sacrament of penance confers grace without any pious exertion on the part of those receiving it, something that the Church of God has never taught or ever accepted. Falsely also do they assert that contrition is extorted and forced, and not free and voluntary.

The document clearly mentions that perfect contrition is sufficient criteria for the absolution of sins. Your question deals with the fate of a person who confesses with heartfelt sorrow. Clearly his attitude comes under perfect contrition and he is worthy of absolution irrespective of the words used by the priest while performing the sacrament of absolution.