What is the Islamic legal ruling concerning the ownership of a house whose value has increased, and does it belong to the borrower (the husband) or to the lenders (the wife's family) after they differed over the nature of the amount (was it a loan or a gift)?

The obligation regarding a loan is to return its equivalent. The lender has no right to demand a share in the house that was bought with the loan money, because a loan entails the borrower's ownership of the money. If it is established that the lenders gifted the money to the wife, then their right to demand it is forfeited and transferred to the wife; she may demand it or waive it. In any case, they have no right to demand the house.