The expression predates that
These may not be the oldest examples (especially with so many variations of the expression), but these examples predate the radio entirely and even nearly predate Grape Nuts itself.
Saints Herald, 1887:
We commend the reading of the following; it contains a deal of good sense:
"EAT BEFORE YOU DRINK.
"A large proportion of intemperance in the use of stimulants," philosophized a physician, "may be laid to the light breakfasts eaten by most people. Breakfast is the most important meal of the day and sufficient importance is not attached to it in the majority of households."
That was reprinted in Good Housekeeping, 1889.
Donahoe's Magazine, 1897:
The breakfast is the most important meal of the day, and the thought and care bestowed upon its preparation will do much to make the day happy.
But opinions at the time varied
Nobody has ever thought that lunch was important (except maybe the French). Dinner, on the other hand...
The Illustrated Oarsman's Manual (1874?) says
Dinner, whether taken in the middle of the day or late in the afternoon, is the most important meal of the day.
When did it become popular?
Google Ngrams shows that growth of the expression was exponential, but 1944 was on the long tail of that. It saw a slight boost in the 1970s and took off in the 1990s. I didn't have time to sift through all the timeframe, but a cursory glance shows that it's the motto of several breakfast companies, such as Ovaltine. Kellogg's went a step further and funded research so they could push that conclusion.