inedible: thin PVC pipe (the size used inside the house to carry the water directly to the sink)
then just wrap the fondant around it.
edible: thick candy sticks glued together (or if know how make your own)
http://t0.gstatic.com/images?q=tbn:TlToD_s0BMVYqM:http://ep.yimg.com/ca/I/candywarehouse_2100_134311533&t=1
I would just do fondant and gumpaste for the flute. I still have my old flute, and it's only less than an inch in diameter (but you're in the UK, so 2 cm or less). It doesn't seem necessary to make anything other than a thick rope for the main structure - even for a life-sized model.
Is the intended recipient actually a flutist? And if so, do you have any idea of their playing level (i.e. beginner, intermediate, or professional)? If they're a relatively serious player, do they have a favorite composer or piece of music, or something they've recently worked on? Do you know the model/brand of their instrument, or have a photo of it? Musicians tend to develop somewhat intimate relationships with their instruments (I speak from my own experience), so some thoughtful detail in that area could score you some nice ooh-and-ah points.
It could be as simple or intricate as you choose to make it. I wish I had this cake to make! I was once a serious flutist, so if you have any specific questions, I'd be happy to help. 
Is the intended recipient actually a flutist? And if so, do you have any idea of their playing level (i.e. beginner, intermediate, or professional)? If they're a relatively serious player, do they have a favorite composer or piece of music, or something they've recently worked on? Do you know the model/brand of their instrument, or have a photo of it? Musicians tend to develop somewhat intimate relationships with their instruments (I speak from my own experience), so some thoughtful detail in that area could score you some nice ooh-and-ah points.
YummyChoo (that makes me giggle), SugarFiend's idea is FANTASTIC! I also am a flutist (music teacher by day, caker by night) and I would be blown away if someone took the time to find out one of my favorite pieces to use in my cake design. Total bonus points! See if maybe there is a sibling that you could talk to. I know my sister would be able to give out the details you would need to pull this off. Best of luck, CAN'T wait to see the results!
I'd probably take an edible image approach to this. I'd find a good photo or illustration of a flute and print it out twice on icing sheets. The first print, I would adhere to some gumpaste and cut out with an Xacto. I'd dry it over a pvc pipe so it would have something of an accurate form. Then, I'd take the second printout, cut out the keys and affix them to more gumpaste. Those would be applied over the original shaped flute to give it more realism. Maybe a light dusting of silver luster dust.
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