In the third episode of Game of Thrones season 6, Oathbreaker, only four people are executed by the Lord Commander. Further, they were executed by hanging, unlike, for example, Lord Slynt.
Why only these four, and why hanging? Surely the Lord Commander has enough stamina to swing a sword four times! And surely, the youngest of those is no more culpable than the others involved in assassinating Jon Snow.
Can't say why only those four. Two possible reasons is that:
They died when the Free Folk attacked Castle Black at the end of s06e02 "Home".
No one could identify who they were.
Obviously, if they were around to be found, they would be hanging as well.
As for the hanging vs beheading, hanging is the de facto method of execution for traitors, which those four were. They broke their oath (Hence the title "Oathbreaker"). Ser Alliser himself explains in an interview:
Vulture.com: And a brutal one. You might have expected Jon Snow to behead them, as he did Janos Slynt, but to hang them? Compare that to Ned Stark's execution — that quick, clean death now seems to be the mercy Joffrey said it was.
Owen Teale, Ser Alliser's Actor: I asked them, "Why isn't there a bigger drop, when you hang them?" And they said, "Because if it's a bigger drop, you would die instantly. The whip action would snap your neck, and that's it. You would lose consciousness immediately." But this was actually more cruel, more of a punishment, because your feet almost touch the ground, you haven't fallen that far, so there's still some life in you and your body can't help but fight. I said, "That's really gruesome."
snipped
As I remember, the speech that I give to Jon Snow, explaining the way I've lived my life — "I fought, I lost, and now I will rest" — he seems at peace about that. But when the moment comes, and you're dangling, you can't help but fight to live. It's important to see them suffer before they actually die, because as Jon Snow sees it it's insurrection. It's treason, what they've committed. It's important to see them suffer, as a message to anybody else who might be thinking of doing the same.
So basically, beheading is the "nice" execution method in Game of Thrones, while hanging is the "not so nice" method. And Traitors/Treason get the "not so nice" method. While you can argue that beheading is enough to make people think twice, something as cruel as a slow painful death of asphyxiation via hanging (instead of neck snapping) may be a better deterrent.
Examples of beheading being Ned Stark. In both manners. Ned is not the type of man that will needlessly make a man suffer. He uses a quick beheading because he has to do it, but he doesn't need to be malicious. And Ned's own death, via beheading, was a "gift" of "mercy" by king bastard Joffery to his intended, Sansa. A quick painless death.
Certainly this is speculation as the Books haven't gotten this far. I suspect they were hanged because there were many, and much easier to hang them all at once than try to behead them all. In addition, when Lord Rickard Karstark killed the Lannister prisoners, he was beheaded, but those that assisted him were hanged...perhaps beheading is saved for Lords (As you referred, Slynt was a lord). Ser Allister was just a Knight, though you could probably make the argument he should have been beheaded.
6 people stabbed Jon Snow, 4 were hanged. It is unclear, but 2 people were killed when the Wildlings came into Castle Black to keep Ser Allister from Killing Ser Davos and burning Jon's body. Perhaps those were the other 2.
Ned's/Jon's words concerning execution were:
"The man who passes the sentence should swing the sword."
He used a sword swing to sever the rope that triggered the execution of the four guys, so I'd say he technically stuck to his words, barely.
Also, Jon is clearly distressed during this episode, as someone who had just been stabbed six times and reincarnated. You can clearly see this from his actions, and the fact he abandons the watch at the end of the episode. Therefore perhaps he has undergone some drastic change in values, so sticking to his original values of beheading execution was probably not at the top of his to do list. The fact he leaves immediately after the execution supports this.
The blood of the First Men still flows in the veins of the Starks, and we hold to the belief that the man who passes the sentence should swing the sword. If you would take a man’s life, you owe it to him to look into his eyes and hear his final words. And if you cannot bear to do that, then perhaps the man does not deserve to die.". Jon Snow did all of that (except literally swing the sword).