Any Manchester United fan, any follower of English football, and most followers of football in general will have some awareness about United's pre-Ferguson history, and in large this is to do with Sir Matt Busby and the Busby Babes. If you don't know who that is, and who the Busby Babes are, you might want to wikipedia the entire thing, it's much simpler.

Now, addressing the question,
Newton Heath F.C, is what Man United were called from their inception in 1878, till 1902 when they were rechristened as Manchester United.
And after the name was changed the old nickname of ' The Heathens ' seemed redundant and irrelevant, so they started being known as simply, 'United'.
Close to the end of World War II, Matt Busby was appointed as manager of United. In 1952, after a wait of 41 years, he delivered the League title to the Red of Manchester, and had also won an FA Cup in those years. But the side was ageing and some of the club's promising youth prospects were being drafted in to the first team, gradually.

This influx of young blood into the starting XI prompted the nickname ' The Busby Babes' from the English media and footballing fraternity.
And after the tragedy of 1958, when many of those Busby Babes were killed
in that air crash after a failed take off at Munich airport, Sir Matt decided that the nickname was inappropriate as so many of the original Busby Babes were no more.
Co-incidentally, the rugby club of Salford [in Manchester ; this club had even used Man United's stadium on occasion], which is now called Salford Red Devils, were on top of their sport as well, and having toured France wearing red shirts and doing remarkably well, they were dubbed ' Les Diables Rouges'
by the French press, and as you can probably guess, it translates to ' The Red Devils'.
Sir Matt, liking the sound of it, declared that United should also be called the same, and this also helped the club change its makeup from being a side consisting mainly of junior academy kids, to a side that mixed the best of those kids with more experienced professionals, not to mention overcoming some of the grief of the 1958 crash.
The club than began incorporating the devil logo into scarves and match programmes, and by 1970, the first version of the current club crest of United, with a devil holding a pitchfork in the centre of the badge.
So yes, this is the the widely accepted accurate account of how the nickname and the crest came to being.