I found interesting information for your question. There are several web sites where you can find answers from people working on research about this syndrome. Here you have a web site:

peacehealth

According to what I read from those web sites, the physical appearance of old age for the Hutchinson-Gilford Progeria syndrome (HGP syndrome) is related to several causes. At approximately nine to twenty months of age, children affected by this syndrome experience growth delays, resulting in short stature and low weight. By the second year of life, the scalp hair, eyebrows, and eyelashes are lost (alopecia), and the scalp hair may be replaced by small, downy, white or blond hairs. Additional characteristic features include generalized atherosclerosis (a condition in which fatty material collects along the walls of arteries) cardiovascular disease and stroke. They also experience loss of the layer of fat beneath the skin, a fact that produces wrinkles.

From the next web site, at wikipedia, you can read the text:

wikipedia_HGPS

"A 2006 report in Nature said progeria may be a de novo dominant trait. It develops during cell division in a newly conceived child or in the gametes of one of the parents. It is caused by mutations in a LMNA (Lamin A protein) gene on chromosome 1.

Nuclear lamina is a protein scaffold around the edge of the nucleus that helps organize nuclear processes such as DNA and RNA synthesis.

Prelamin A contains a CAAX box at the C-terminus of the protein (where C is a cysteine and A is any aliphatic amino base. This ensures that the cysteine is farnesylated, and this allows Prelamin A to bind membranes, specifically the nuclear membrane. After Prelamin A has been localized to the cell nuclear membrane the C-terminal amino acids, including the farnesylated cysteine, are cleaved off by a specific protease. The resulting protein is now Lamin A, is no longer membrane-bound and carries out functions inside the nucleus. In HGPS the recognition site that the enzyme requires for the cleavage of Prelamin A to Lamin A is mutated. Lamin A cannot be produced and Prelamin A builds up on the nuclear membrane, causing a characteristic nuclear blebbing.[2] This results in the premature aging symptoms of progeria, although the mechanism connecting the misshapen nucleus to the symptoms is not known."