Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"?
"Hurray!" cried Pippin, springing up. "Here is our noble cousin! Make way for Frodo, Lord of the Ring!"
"Hush!" said Gandalf from the shadows at the back of the porch. "Evil things do not come into this valley; but all the same we should not name them. The Lord of the Ring is not Frodo, but the master of the Dark Tower of Mordor, whose power is again stretching out over the world. We are sitting in a fortress. Outside it is getting dark."
The Fellowship of the Ring, chapter Many Meetings
"[E]ven if we could [hide the Ring], soon or late the Lord of the Rings would learn of its hiding place and would bend all his power towards it."
Glorfindel, in The Fellowship of the Ring, chapter The Council of Elrond
Both comments show that even while not wielding the One Ring, Sauron is still the Lord of the Rings.
The full title of the Red Book of Westmarch is:
The Downfall of the Lord of the Rings and the Return of the King
pointing clearly to the Lord of the Rings being Sauron.
Lord of the Rings, plural. Sauron is the only being who could have this title, because he is not just the master of the one ring, he also maintains control of the other rings of power that he had planned to slave to the one ring. The elves thwarted his plans for them, and the dwarves were resistant. But he definitely controls the nine human rings.
Really that title refers to Sauron in more ways than one. He was the one who gave the secrets of ring making to Celebrimbor. He was involved with the making of all but three of them. It's really a literal description as much as a title.