What is the ruling regarding the phone in my possession? Is it considered forbidden (haram) or permissible (halal) for me to keep it, after the owner's mother refused to receive it and denied her son's ownership of it, knowing that I paid a sum of money to retrieve it from those who had impounded it?

If you did not spend money to free the two unjustly imprisoned individuals, then it is not permissible for you to take money in return for your efforts to free them, because repelling injustice is an obligation. As for the money you spent to recover his possessions, it is permissible for you to demand it back. If he is not willing to repay what you spent, then it is permissible for you to take the amount of your right from his possessions and return the remainder to him. This falls under the category of "obtaining one's right."