Call your insurance agent if you have one. If not, or also, call your local building inspector. It matters less what we think of what you have, and plenty to either or both of them. If the inside isn't a lot tidier than the outside, the nearest reliable liner installer guys are usually up on local requirements.
Also depends a bit on what vents through the chimney. Solid fuel heaters usually cause bigger temp fluctuations than similar gas or oil devices. Possibility of freezing between heating cycles wants something to minimize moisture absorption in veneers or facings.
A lot of small, post-war homes around here have metal weatherproof housings or shrouds with brick-looking paint jobs. Some were built with metal chimneys from day one, others a refit for sale or for insurance specs. I've seen both types go up in smoke after wood heaters were added to tiny homes built with wall heaters vs larger, more modern forced air units. Flue fires are no fun to stand and watch while fire vols are still on their way.