My snake's neck is a little swollen too.
It used to be really thin but then after he got sick a couple years ago it got a little bigger.
I was a little concerned but I've seen tons of pictures of snakes with thicker, seemingly swollen necks. All hatchlings seem to have extremely thin necks, so I concluded that three possible answers might include that it's just his age, or his neck may be swollen from the endless sneezing or the regurg. Or he's just getting fat. (This occurred to me last night. My snake doesn't eat that much, no more than he always has since he was young. But I've heard a lot about overweight snakes. And then I wondered how can a snake be overweight? Have you ever seen a snake with FAT on it? Snakes always seem like pure muscle.)
Actually I'm still concerned and every day for quite a while, I've considered posting a question and pictures about it. But I haven't gotten around to it, and since there are a ton of other snakes out there whose necks look the same (I was even going through the old corn snake manual just last night, and half the snakes there have big necks too) I'm not sure that it's even a problem.
He still looks completely healthy, his behavior is normal and he hasn't turned down a single mouse since he was sick.
However my snake's throat is not nearly as dramatic as yours seems to be. It's likely not even the same thing.
I guess after saying that, I don't know why I posted this. Maybe someone would know why some snakes look to have slightly swollen necks. I haven't noticed a relevence between shedding and neck size. I guess that could be it but he definitely doesnt look anywhere near shedding.