This little phrase has several meanings - an advance warning - being wide awake and alert - being the head of - a type is display screen.
Several of the meanings of this phrase allude to holding one's head up high and concentrating on what one is doing. Residents of the USA will have come across heads up as an advance warning; for example, 'The boss was coming. Jim gave us a heads up to get on with some work'. That usage is fairly recent and hasn't yet become commonplace outside America. The first example that I can find is from The Washington Post, August 1979:
"In a message characterized as a 'heads up alert', intelligence officials warned ... that Arab diplomats had suggested that Ambassador Andrew Young meet with a Palestine Liberation Organization official."
This meaning is a modification of an earlier meaning of heads up, that is, look alive! - watch out. The earliest I can find for that is also from the Washington Post - from November 1914:
"Heads up". A baseball and football term signifying alertness, action,
Many of the citations of this meaning refer to sports, especially baseball and American football. There's a style of play in baseball that is called 'Heads up baseball'; for example, from the New York Times, June 1924, we have:
"The entire Tiger team played heads-up ball and closed the year with nineteen victories and four defeats."
That sense of holding one's head up and being alert and energetic is also expressed in this item from Collier's Illustrated Weekly, 1914:
"Heads up, you guys"... We ain't licked yet."
Heads-up (or head up) display screen began to be used in aircraft in the 1960s. These project images and information on to the aircraft windscreen and can be viewed by the pilot while still keeping half an eye on the flightpath ahead - so long as his head is up of course. The first citations of that are from an advert in the New York Times, 1967, for a "Heads-up display", and in this definition from the Toronto Globe & Mail, 1978:
"There are experiments with what pilots call a ‘heads-up display’ that projects vital information on to the windshield."
Another meaning, which has no connection with one's own head is when heads up is used as a verbal phrase; for example, "We have a new boss. He heads up the whole organization."
See other phrases that were coined in the USA.