The plane was a last resort because enough shielding would be too heavy, so the crew knew they would not survive the mission. I'm thinking the plane might have been a "flying wing" shape
Is it possible you're thinking of the novel Steam Bird by Hilbert Shenck? This from 1988 not the 1950s, but you did put a question mark after the date and this does appear the sort of pulp book that would be at home in the 50s.
This is a hard book to get information on - I suspect it is enjoying well deserved obscurity. I found a reference to it in an anthology The Orbit Science Fiction Yearbook 2 from 1989. In the end notes the anthology mentions:
It is a grown person’s guide; and Steam Bird (Tor) is a grown person’s cap in the air. Highly silly—a steam-driven nuclear bomber takes off from Maine and circles the world for locomotive hobbyists and democracy—and remarkably affectionate, “Steam Bird,” the novel-length tale which takes up most of the volume, glows with an innocence only available to real folk on holiday.
I've tried Googling for the book but found little more than that it is indeed about a steam powered bomber and that it nearly triggers WWIII so there is apparently a reference to world war three.