I know that many clone troopers became Imperial stormtroopers, but for how long? Do we have any information on the last of them?
It depends which clones you're talking about.
According to the video game Star Wars: Battlefront II, Fett's clones had almost completely died out during the Kamino uprising. The Fett clones struggled to integrate into the more general army and, according to Survivor's Quest, by 22 ABY the remainder are thought to have died or deserted.
If you include non-Fett clones, you can include those created after the Spaarti took over cloning on Coruscant. They had a process which could generate matured clones in a year. However, their clones ended up being inferior and deformed, leading to mental instability, lack of fighting power, and as such resulted in a high casualty rate. These probably died out or deserted at the same time as the last Fett clones.
Currently, there isn't a Disney canon answer, as we don't know the makeup of the Imperial Stormtrooper Corps at the time of the original trilogy. But as far as Legends canon goes, we do have a couple answers:
As Polynomial notes in his answer, the last of the Fett clones were created about the time of the Kamino Uprising in 12 BBY, and these "anti-troopers" were all destroyed. No more new Fett clones were created, and as he pointed out, these eventually died or deserted. Given the accelerated aging, not many clones would be alive even 40-50 years after their aging.
However, one exception to this is Jaing Skirata. He was one of the first Fett Clones, a Null ARC trooper who was cloned in 32 BBY, and was able to stop the rapid aging using stolen Kaminoan research. Still alive at the time of the Legacy of the Force series, he was in reality over 70 years old, but biologically 140. Using this research, several other clones were also able to extend their lifespan, and though we are not told of their names, it is likely that these were the other Null ARC Troopers. However, it is almost assured that these few were the only Fett clones still alive around 40 ABY (and thus the last ones in the galaxy), as the 'unfixed' clones would have died of biological age long before.
Around 40 ABY (towards the end of Legends canon), we do see some research still being done on cloning (by a Kaminoan, no less), but not to the level of the clone troopers.
For an answer from canon, the best information we currently have is from the canon novel Tarkin. It includes a scene where a group of stormtroopers are led by a Fett clone sergeant but all the other troopers are non-clone recruits. Here is the relevant quote (the stormtroopers are loading Darth Vader's meditation chamber onto Tarkin's ship):
When the stormtrooper operating the equipment accidentally allowed the flattened sphere to bang against the edge of the cargo hold’s retracted hatch, Vader stamped forward with his gloved hands clenched.
“I warned you to be careful!” he shouted up at the trooper.
“My apologies, Lord Vader. Wind shear from—”
“Excuses won’t suffice, Sergeant Crest,” Vader cut him off. “Perhaps you are aging too quickly to remain on active duty.”
Tarkin couldn’t make sense of the remark until he realized that Crest’s was a face he had seen countless times during the war—the face of an original Kamino clone trooper. The bare-headed others comprising Vader’s squad were human regulars who had enlisted after the war.
The novel takes place about 5 years after the end of the Clone Wars (14 BBY), so clone production evidently stopped at the end of the Clone Wars and within 5 years the clones were becoming too old to be useful as soldiers (due to their age acceleration). The Empire likely phased out the last of the Fett clones within a few years of this event. For all we know, Sergeant Crest was the last clone stormtrooper.
If you are looking for the oldest surviving clone in the new Disney canon that isn't necessarily a stormtrooper or still serving the Empire, then Captains Rex and Gregor, and Commander Wolffe and are certainly in the running.
In the Star Wars Rebels episode "The Lost Commanders" (S02E01), the crew of the Ghost is sent to find a "great military leader" from the Clone Wars, who turns out to be Rex.
Given that Rebels is set 4BBY and given the apparent ages of Rex, Wolffe, and Gregor (biologically, not necessarily chronologically), it would seem likely that the Empire would not have too many Clone Wars clones left at this point in time and they could be among the last survivors.