What is the ruling on buying a car for a certain sum, then selling it to partners for a higher price by installments; is there any suspicion of unlawfulness in that, and what is the legitimate way out if there is?

The summary of what you mentioned is that your uncle appointed you as his agent to manage a sum of money. You then used this money to purchase a car in partnership with two other individuals, resulting in your uncle owning six shares of the car and each of you owning one share. Subsequently, you sold your uncle's shares to yourself and your two partners. This sale is generally permissible. However, since you are one of the buyers, scholars have restricted the permissibility of an agent selling to themselves unless the principal has authorized them to do so, or if there is no one else interested in buying the item. Therefore, if your uncle is content with the sale to yourself, or if there were no other interested buyers for the car, then what you did is permissible.