Per Humphrey Carpenter's biography of Tolkien (which is the source of the above quote; read it several months back), I can't remember there being specific aging or health concerns beyond the general wear and tear our bodies experience throughout their lives. Him and Edith, as that quote indicates, moved to Bournemouth as their popularity became too much for them to handle, and lived pretty happy, simple lives.

Dunno how much you know about the couple in general, but Ronald and Edith were very deeply in love throughout their lives; while Tolkien was a kid, he was forbidden by his guardian for a while to speak to her, so their romance even has a sort of fairy-tale, forbidden aspect to it. As soon as he turned old enough to write her, was no longer forbidden to speak to her, he wrote her a letter immediately at like midnight. I think their care for one another is more related to the intense, lifelong devotion they had for one another than as a result of actual health concerns, that's the primary point I'm trying to make.

Still, though, this is what I could find to suggest anything health-wise, skimming through the book again, same source, Carpenter: "In some ways, [Tolkien] found old age deeply distressing, while in other respects it brought out the best of him. He was saddened by the consciousness of waning powers, and wrote in 1965: 'I find it difficult to work - beginning to feel old and the fire dying down.' Occasionally this plunged him into despair, and in his later years he was particularly prone to the gloom that had always characterized his life; the very sense of retirement and withdrawal was sufficient to bring out this side of his nature.

Very feasibly, these could be some of the concerns remarked upon.

Shortly thereafter is the remark that their new home on Sanfield Road "was two miles from the center of Oxford, and the nearest bus-stop was some distance away, further than Edith was able to walk with ease," which you could take to indicate rheumatism or arthritis, possibly.

Some evidence to hint it was less due to health ailments and more a remark on their "absurd" degree of care for one another: "Nor would he dine away from home unless Priscilla [Their daughter] or a friend was able to keep Edith company for the evening. Concern for her well-being always took precedence with him."

Another, which I think indicates that Edith did have her facilities well into old age: "When the strain of keeping house became too much for Edith, it was easy to book their usual rooms... to take them down to Bournemouth. At the Miramar, Edith would soon recover much of her strength, not to say her good spirits..."

Edith herself died of an inflamed gall-bladder, it says, and Tolkien passed away shortly after her. The cause of death being an inflamed gall-bladder, not cancer or some other wasting disease, leads me to believe that the death was more a result of aging than of a long-standing health concern. As for Tolkien, he passed away as a result of an acute bleeding gastric ulcer that developed into a chest infection.