Nah... It predates Churchill by a century.
I believe that the very first person who said that was Francois Guizot (1787-1874): "Not to be a republican at twenty is proof of want of heart; to be one at thirty is proof of want of head." Georges Clemenceau was very fond of the quote and repeated it himself.
Variations on this quotation are often attributed to King Oscar II of Sweden, Disraeli, Shaw, Churchill, and Bertrand Russell, among others.
The sentiment was very popular in the communist country where I grew up--everybody was constantly quoting it and winking knowingly.

Of course, we substituted "communist" or "socialist" for Guizot's original "republican." Not to be confused with our Republican party...
If You Are Not a Liberal at 25, You Have No Heart. If You Are Not a Conservative at 35 You Have No Brain | Quote Investigator