There are already excellent answers to the bulk of the question, but perhaps not to "Were they [the Ring-Bearers] mortal once again because the Ring's power had been broken by its destruction, or the magic of the Undying Lands, or did the Valar themselves reverse it?"
Tolkien does answer this. When Bilbo is talking with Gandalf, he says:
'I am old, Gandalf. I don't look it, but I am beginning to feel it in my heart of hearts. Well-preserved indeed!' he snorted. 'Why, I feel all thin, sort of stretched, if you know what I mean: like butter that has been scraped over too much bread.
(Some useful dates: Bilbo was born in 2890. He acquired the Ring in 2941 at age 50. He gives up the Ring in 3001 at age 111. Frodo sees Bilbo in Rivendell in 3018 when Bilbo is 128 and he is 129 when the Ring is destroyed. He leaves for Aman at age 133. He possessed the Ring for 60 years.)
So when Bilbo is 111 (comparable to mid-late 80s for a human) he looks fifty, but feels "stretched" and "like butter that has been scraped over too much bread". The Ring confers unending life but not more life. The "immortality" of the Ring is a poisoned chalice.
Bilbo lives in outward vigor and youth until he gives up the Ring. When Frodo sees him in Rivendell (Bilbo is around 130 -- like one of us at 100), Bilbo is depicted as older, but not yet showing his full age. Bilbo says:
I have thought several times of going back to Hobbiton for it; but I am getting old, and they would not let me: Gandalf and Elrond, I mean. They seemed to think that the Enemy was looking high and low for me, and would make mincemeat of me, if he caught me tottering about in the Wild.
When Bilbo asks to hold the Ring, Frodo sees him as
a little wrinkled creature with a hungry face and bony groping hands.
Now the emotional part of this is false and due to the Ring, but note that Bilbo is objectively described as "wrinkled" with "bony" hands. Also, Bilbo is usually described as the "old Hobbit" while in Rivendell.
So while Bilbo no longer has the Ring, but the Ring still exists, he began to age normally or perhaps a bit faster than that. When Frodo meets him in Rivendell, Bilbo is 128 years old (human ~100) but held the Ring for 60 years and looks like he is getting old, but is still healthy. This fits very well with him show a hobbit-age of just the years he was without the Ring: 50+18 or about 68. It is consistant with the book that the years Bilbo held the Ring do not affect his age as long as the Ring exists.
But once the Ring has been destroyed, Frodo returns to Bilbo in Rivendell:
[Frodo] found him all alone in his little room. It was littered with papers and pens and pencils; but Bilbo was sitting in a chair before a small bright fire. He looked very old, but peaceful, and sleepy.
He's now fully showing his great age -- he's a centenarian by our standards and showing every year of it.
Bottom line: Once Bilbo gave up the Ring he started ageing again normally. When the Ring was destroyed, his full age quickly returned. By the time he left for Valinor, he was a very old Hobbit -- the oldest for which there was a record -- but not yet into supernatural territory.
With the Ring destroyed, the various Ring-Bearers were their true ages and the Ring's physical effects were gone. They were mortal.