You'd have to separate the legal holding of the title from from the right to hold the title *and* from the right to wield the power of the title. But defining a process is hard. IMHO you lose the right to the title immediately but congress need to invoke formal disqualification and transfer

So similar to your other reply, while the president holds the title legally until congress acts, at first somebody like maybe VP or speaker claims the president has lost right to the title (starting process of impeachment), and if it escalates they declare the right suspended (this will get messy)