The mortar is cracked almost from top to bottom. The pictures don’t show it, but there is also some separation at the wall, about a 1/2". I’ve recommended that they have a mason or builder have a look, dig down and check the footing. Is it a simple fix or a tear-down?
Like William stated, anything can be repaired with time and money.
Looking at your pictures, it appears to have a 12"x16" flue with 6"x12"x8" CMU surround.
Since we can not see the top of the chimney cap, I can only speculate that there is an un-caulked gap between the flue and the CMU. Water infiltration and freezing might be the cause, but it could also be thermal expansion from the way it was constructed.
Here are a couple of links that may be useful in assessing the cause if that is what you are looking for.
From an inspection point of view.
Reccommend a Chimney Mason Contractor to assess and repair. http://inspect-ny.com/chimneys/Chimney_Outdoor_Inspect.htm
I believe that is a knock down and do it all again deal, the blockwork should never have been left unfinished as moisture quickly wicks into CMU, I suspect that moisture in the CMU has repeatedly frozen which is a large part of the problem.
Try this–Unsealed/unpainted chimney cmu’s are subject to water in-migration with possible freeze/thaw conditions as a result. Recommend a certified masonry specialist
or W.E.T.T. technician (at least) further assess the condition.
Typical to see that there is no rain cap. Especially in our wet coast weather.