Does a mental disorder exist where people think someone else is sick? Is there a known disorder where the subject thinks someone else or other people around them are sick or dying, say exo-hypochondria?

In this case I am thinking of a wife pathologically convinced that her husband is dying or seriously ill of some unspecified condition and that he knows this and is deliberately concealing it from her.
There is a condition called hypochondriases by proxy (e.g., Moreira &amp; Moreira, 1999). It is also referred to as vulnerable child syndrome (Pacurar et al., 2015). The parent, particularly if (s)he is inexperienced or unsupported, might see symptoms and illness in a child where in fact there are none. For instance, the parent might be convinced that a common cold is a respiratory infection, or that a simple headache is a brain tumor. In such a scenario, the parent might exaggerate the child's symptoms in order to get the physician to take her fears seriously. This would then be akin to a "cry for help" rather than an intentional desire to malinger (Mart, 1999).

Hypochondriases by proxy is reported to be related to Munchausen syndrome and Munchausen by proxy syndrome, which belong belong to a group of diseases called factitious disorders. In these disorders, a healthy person feigns to suffer from disease, illness, or psychological issues to draw attention or sympathy, or they can project a disease or illness upon to another person, asking or requiring medical intervention for them, respectively 
(Pacurar et al., 2015).

References
- Mart, Am J Forensic Psychol (1999); 17(1): 69
- Moreira &amp; Moreira, J Pediatr (Rio J) (1999); 75(5): 373-6
- Pacurar, Romanian J Pediatrics (2015); LXIV(3): 252-8