People often ask me, do miss turtles are buddy or mate that has moved away or died. My answer used to be, "probably", but over the many years I've been with animals, I have to change that to a positive "definitely."
After all, emotions are formed in the oldest parts of our brains. And since animals can feel fear, attachment, joy, anticipation, and disappointment, there is no reason to exclude sadness and grief.
I used to be very skeptical when people attributed human-like emotions to turtles. Turtles in captivity live very differently than turtles in the wild, and the emotions that come into play are probably different, too.
It may be your turtle loves you more for the tasty treats you provide every morning than for your charming personality. Or it will anticipate the enjoyable scratching and petting over a heart-felt verbal greeting. But that's a difference in circumstance and trigger, not emotion felt. Some of my red-eared sliders come up to the edge of the pond specifically to be petted. I've tested this and threw food into the water which they ignored, but they wanted the petting. From me specifically. Because they won't let other people that they know pet them. I feel loved.
So, there is no reason to assume that turtles don't mourn when a buddy they've spent a large portion of their life with moves or passes away. And there is a lot of indication, that they do. In the wild, turtles may bask with each other on the same log, but they are not housed in close quarters for decades, court the same partners, so I would guess it's a little different in the wild than in captivity.
I had a pair of Reeves turtles together for several years. I had to separate them when the males advances became too rough. The female seemed clearly relieved to be rid of him. And he kept looking for her everywhere in his new surroundings for quite some time.
So, yes, I think turtles, and other reptiles, miss their buddies when they get separated, and they go through a period of mourning.
Should you get them a new buddy?
That depends. You are taking your chances, of course, of them getting along with the new buddy. But you are also giving them back something they've gotten used to, which is company. It's a good idea that any new buddy be of similar size and age, and if possible, temperament. And be prepared for it to not work out, just in case.
Disclaimer: All of the above is subjective and based on my personal experiences not scientific research. However, scientific research is starting to lean towards confirming the above.