Is saying "I will pray the prayers on time" or "at the beginning of their time" considered a vow that must be fulfilled, given the difficulty in doing so due to studies? And is tribulation in religion and worship an indication of God's wrath and His not loving the person?

This is not considered a vow for two reasons: The first is the doubt regarding the wording that was uttered. Performing prayers on time is an obligation and cannot be a vow. As for praying at the beginning of its time, although it is valid for a vow, the doubt keeps the conscience clear. The second is that a vow requires a specific formulation, and your statement "I will pray" is merely a resolve that does not constitute a vow unless accompanied by a firm intention. However, a Muslim should not say anything with their tongue except what they intend to do. Delaying prayer from its early time is a shortcoming, and delaying it beyond its prescribed time is a grave sin, requiring repentance. One must not surrender to religious compulsions (waswasa) in matters of faith. As for trials in religion, they are not evidence of Allah's anger; rather, they may be evidence of His love, as the Prophet, peace and blessings be upon him, said: "Indeed, the greatness of the reward is commensurate with the greatness of the affliction. And indeed, if Allah loves a people, He afflicts them. Whoever is content, for him is contentment, and whoever is displeased, for him is displeasure."