I heard a children's book read on the radio in the 1970s.  The plot
was about a family where one by one the mom, dad, and child came down
with a bad cold (?flu) and were cared for eventually by gramma, who
finally became ill herself and was cared for lovingly in return by her
family.  She said something like "it was the best, bad cold she ever
had".

Hello.

The book was probably "Tell Me a Mitzi" (1970) by Lore Segal.

See this description from a school reading list:

"Tell Me a Mitzi
 In three unique, comical adventures, Mitzi wakes her brother Jacob
early to try to take him alone to her grandparents' house, Mitzi's
mother takes care of the whole family when one by one they come down
with colds, and, out for a walk with Mitzi and their father, Jacob
stubbornly insists that he wants to see a parade."
source:
Rodeph Sholom School: SUMMER LISTENING
http://www.rodephsholomschool.org/library/kind.pdf

"Three funny tales of a city family, mixing fantasy with reality: a
mad trip to Grandma`s house, coming down with a cold, and a meeting
with the President."
http://www.buy-books.us/894326.htm

"Three household adventures in the life of Mitzi include an intended
trip to grandmother's, sharing a family cold,"
http://www.alibris.com/search/search.cfm?qwork=6598285&qsort=r

---------
search strategy:
"one by one" "with colds" book

I hope this is the book that you had in mind.  If not, please let me
know via the "request clarification" feature. Thanks.

Great, quick answer.  I just ordered and received the book and it was
exactly what I was looking for.  Thanks for the brilliantly successful
answer!