Does the wife's guardian have the right to prevent her from her husband and refuse to complete the marriage without a legitimate reason?

If the legitimate marriage contract with your paternal cousin has been concluded, she has become your wife, and your paternal uncle must hand her over to her husband after he has paid the prompt dower. He is not permitted to be stubborn or use her studies as an excuse. Your paternal uncle does not have the right to order you to divorce her, and you are not obliged to comply unless you wish to. Continue to seek mediation from people of goodwill, and if they are unsuccessful, it might be better to divorce her or refrain from divorcing her until she ransoms herself from you with compensation (khul'), and the matter can be escalated to the Sharia (definition) court in case of dispute.