There is a proof of cross-immunity with MERS, SARS, and more difficult to study common colds types (because people, in general, don't test the strain of common cold coronaviruses).
The latest strong believe in some scientists (and a very logical conclusion), is that your T-cells, which memorize pathogens they fought trough your lifetime, can instruct and mobilize your immune system very fast if they were exposed to a similar virus.
That also will make a difference in the fatalities, in different countries and regions, as flu and cold, is very often localized.
By the way, T-cells diminish with age