On the subject of Lords
The British system of Peerage dates back many centuries. When a Lord dies, his son is automatically elevated to his title, regardless of age. For example, Eadwig (House of Wessex) became King at age 15 (955-959 A.D.) when his father's brother King Eadred died in 955 (his father died in 946). King Edward the Martyr took the crown in July 975 at age 13. And King Æthelred the Unready (978) ascended to the crown at age ten. King Henry the 3rd (Henry of Winchester, House of Plantagenet) was crownd in October 1216, less than a month after he turned nine. Henry the 6th (House of Lancaster) was born December 1421 and crowned King in August of 1422, not even making him one year old. Edward the 5th took the crown in 1483 when he was twelve. Edward the 6th took the crown in 1547 at age nine.
Nobles, of course, took their lead from the crown and followed the same rules. If a man held a peerage, his son would succeed to it; if he had no children, his brother would succeed. If he had a single daughter, his son-in-law would inherit the family lands, and usually the Peerage; more complex cases were decided depending on circumstances. Customs changed with time; Earldoms were the first to be hereditary, and three different rules can be traced for the case of an Earl who left no sons and several married daughters. In the thirteenth century, the husband of the eldest daughter inherited the Earldom automatically; in the fifteenth century, the Earldom reverted to the Crown, who might regrant it (often to the eldest son-in-law); in the seventeenth century, it would not be inherited by anybody unless all but one of the daughters died and left no descendants, in which case the remaining daughter (or her heir) would inherit.
The King could assign a "guardian" or "regent" if no family members were available to rule in place of the young child until s/he could take the reins of responsibility.
In the Wizarding world, the King's prerogative would probably be replaced by the Wizengamot.
Harry's Wizard family is dead, and his godfather is "unavailable," so the King, in this case the Wizard's Wizengamot, following the traditions, would have appointed a regent for infant Lord Harry. In addition, only the King, again, the Wizengamot, can set aside the terms of a Will — but those terms must first be disclosed. Again, the rules regarding wills go back over a thousand years. Burying or sealing the Will is flatly illegal for all but the King or Queen. Concealing a Will and preventing its execution is a crime that leads straight to prison.
Thus, Harry Potter immediately became Lord Potter when his parents died. The Wizengamot should have appointed a regent to control his estates until he became old enough to assume direction of his estates himself. And old enough would be when he started making decisions instead of letting his regent do the decision making for him, not based on an arbitrary age limit. The idea that you have to be sixteen or eighteen before you are considered a legal adult is a fiction invented in the 1900's.
If there is some question as to who takes the position of Lord when the current Lord dies, then the contenders are all referred to as Apparent Heirs until the situation is resolved. In Medieval times that meant by whomever could purchase the better army or assassins.