http://www.youtube.com/watch?v=SXfSXi8TT0w
A more complete version.
There are more than one Flute sounds used.
I would venture to say that the opening one is a synthesized sound similar to a pan flute in quality also called in some countries by the name of Syrinx.
Later on in the piece, they use a standard concert flute.
To hear something similar I would research the S. American tune, "El Condor Pasa"
or just see this vid:
http://www.youtube.com/watch?v=CtUZzCe6-bk
The melody doesn't really start until about :35 into the vid and it's not necessarily showing the instrument being played while you are watching.
Pay attention to the vibrato. If I may add, I'd say that they aren't using finger vibrato. (Esp. with pan flutes) it's a breath vibrato on either the pan flutes or the traditional fipple flutes. Finger vibrato "the Softening" as it was called in the baroque period and in some folk music is a bit more subtle as it diminishes the tone when used but also it's a bit less consistent.
http://www.youtube.com/watch?v=k10q2OZpnsc
<Added>from a bit of a search, the synth programmer on the movie was Ian Underwood. Ian Underwood is an interesting individual from what I see in his experiences and I would contend that the "Flute" in the beginning of this song is synth but later it would have added the Flutist from the LSO. Mixing the live with the synth makes the synth sounds more human to most listeners.
"Personnel: James Horner (conductor, keyboards); Choristers of Westminster Abbey (vocals); Tony Hinnegan (kena, whistle); Eric Rigler (Uilleann pipes); Mike Taylor (bodhran, whistle); Ian Underwood (programming); London Symphony Orchestra.
Recorded at EMI Abbey Road, London, England. Includes liner notes by Mel Gibson.
All music written by James Horner."