The mortgage lender does not, never has and never will 'own' any part of your house. You own 100% of it and you always have since the day you bought it (unless, of course, some sort of shared equity scheme was involved when you bought it). You simply have a loan that's secured against the property.
When you die your house will form part of the assets of your estate, with the outstanding mortgage (along with things like credit card debts) forming the liabilities. The beneficiaries of your estate can neither take possession of your the assets or use them to settle the liabilities. It's the task of the executors of your will to balance the books and then to distribute your estate in accordance with the terms of your will. So they'd use the money in your bank account to pay off your debts (including the mortgage) and then dispose of the house in the way that you wanted them to do. If your will says "I give my house to Fred Bloggs" then the title to the property will simply be transferred to him. If your will says that the house should be sold (or there's no specific instruction about it) your executors will be free to sell it straight away or, if they so choose, to wait until house prices have risen.