For everyone but Jon...
...it doesn't really matter what the rules would strictly say about Jon's status:
Castle Black has just been, essentially, invaded by an army of wildlings and a giant. He's their chosen leader. It doesn't really matter if he orders the execution as Lord Commander or conqueror, the result is the same. The rules are as irrelevant at this moment as the fact King Robert wasn't the next in line to the throne when he put himself on it. Varys put it rather well:
Power resides where men believe it resides
Alliser could have protested against this, but chooses not to because he still sees Jon as the Lord Commander, and accepts his defeat. From a transcript:
Alliser: I had a choice, Lord Commander. Betray you or betray the Night's Watch. You brought an army of wildlings into our lands. An army of murderers and raiders. If I had to do it all over knowing where I'd end up, I pray I'd make the right choice again.
Jon: I'm sure you would, Ser Alliser.
Alliser: I fought, I lost. Now I rest. But you, Lord Snow, you'll be fighting their battles forever.
Alliser is very absolute and black-and-white, not the type to quibble about interpretations. He'd probably rather hang than try to get off on a technicality.
As head steward, Bowen Marsh (the first of the four) would be the most likely to raise a technical objection - but strict conformity to the rules is clearly not his focus at this moment:
You... you shouldn't be alive. It's not right.
He's a bit too distracted by the whole "the man I murdered is back from the dead" thing to worry about whether the man he murdered is, technically speaking, overstepping his authority. It'd take a ballsy lawyer to say "You lost the authority to execute me when I murdered you!".
...it's hard to say whether this would bother him. He's always been someone torn between doing what the rules or honour say (e.g. not joining Robb's army), bending honour to do what he thinks is right (e.g. trying to assassinate Mance during negotiations), and just doing what he wants to do (e.g. Ygritte).
But we've got reason to expect undead Jon to be colder and harder than he was before, based on what Beric said about his own returns from the dead back in season 3:
Every time I come back, I'm a bit less. Pieces of you get chipped away.
Also, in the books:
It seems like the longer someone is dead, the colder they come back. A certain lady who was dead a rather long time, who had been rather compassionate, comes back distinctly stone-hearted... Beric was never dead longer than minutes I believe, this lady was dead for days; it stands to reason Jon would be somewhere between the two.
Jon was never entirely above bending the rules when it suited him, and a potentially vengeful undead Jon would seem to be even less likely to worry about such details.
The way he walks straight out with only the slimmest of attempts to justify himself...
My watch is ended
...is a pretty good indication that he worries less about such details. As is one of the very first things he says after coming back:
I did what I thought was right. And I got murdered for it.
Update from episode 4:
Regarding Jon's status at the Wall:
It's confirmed that there's disagreement and confusion about whether Jon is Lord Commander or not: a Brother delivers him a letter, addressing him as Lord Commander, and he hesitates, denies the title, but takes it anyway. There's also debate about whether his vows still apply.
Regarding Jon 2.0's personality compared to the original model:
Based on his conversation with Sansa, he seems to be more lost than vengeful, compared to his old self: possibly less purposeful and driven, rather than more. It certainly seems like he came back "a bit less", but not in the cold, grim, hard and wrathful way I had suspected based on the analogous character from the books.