Is the marriage that took place between the questioner's paternal uncle's daughter and her other paternal cousin valid in Islamic law, knowing that the girl's father had previously agreed to her marriage to the questioner, and that the girl desired to marry the questioner?

Your uncle's action was merely a promise to marry his daughter to you, and fulfilling a promise is commendable. Therefore, he did not wrong you. What happened does not affect the validity of your other cousin's marriage to her. You must accept and submit, for what befell you was by God's permission, and God knows what is good. It is important to note the prohibition of proposing over an existing proposal. If a contract was made for the second suitor, his contract is annulled before consummation, unless he had already consummated the marriage.