Did you see a doctor about the swelling? Or are you just self-treating? I suspect whatever was going on in the first place is probably still causing problems. It's probably best to see a doctor to be sure something very serious isn't being left untreated.
In the interim, for more "common" causes of cold hands such as you describe, possibilities include anemia or a drop in blood platelet levels. If you are not getting the right nutrients, you could cause yourself to become severely anemic. Menstruating women have to be very careful on vegan and vegetarian diets as cutting meats and certain proteins that are good for healthy blood production can bring on anemia. Supplementing these protein sources in a proper way is very important. I suspect this may be what is going on. That means, I don't think you've solved the swollen neck problem and may have created a new one by changing your diet. And by the way, anemia can get worse and there can be complications if left untreated. If your blood remains starved of oxygen caused by the anemia too long you can have heart problems and more.
A much more serious cause could be an undiagnosed heart problem or some type of circulatory problem.
Lastly, it is very possible, whatever ailment was causing the swelling in your neck, could be causing the cold hands too.
Have you considered a Thyroid problem? This can cause swelling in your neck, eyes and many other places if left untreated. A friend and an aunt had your symptoms, left them untreated, and both had complications. My aunt will forever look like Rodney Dangerfield. Her eyes swelled so much she doesn't look like herself anymore. They stick out of her head in a very unnatural way. My friend almost lost her ability to talk forever because she refused surgery on her Thyroid. Thankfully, after much therapy...and what I believe was a miracle, she is now able to speak again and her Thyroid situation is under control. But it was a scary two years.
Throat cancer, and an iodine deficiency could cause this as well. There are so many possibilities. I say go see the doctor and have some tests done. Don't rely on other people's maybes, including my post's suggestions and self-diagnose, if that's what you have been doing. I say get checked and be certain. If necessary, find a specialist or a new doctor.
And if you've already done so, I hope you get better soon. Sounds scary.