Yes, yes they would. Cells need energy and oxygen to operate. The cells that produce the hard keratin we call nails, can’t operate in the absence of fuel any more than any other cell and would die shortly after death. And your nails won’t grow.
They might /appear/ to grow as the skin around them shrivels back though
Death means no more supply of glucose, and therefore no growth of nails. Different cells die at different rates. After the heart stops beating, oxygen supply to the brain is cut off, nerve cells die within three to seven minutes. Kidneys, livers and hearts from donors can be taken within thirty minutes of death and get them into recipients inside six hours. Skin cells, meanwhile, are longer lived. Nails may appear to keep growing, but this is because flesh shrinks as it dries out, skin is pulled back to make the nails appear longer.