Ash is definitely the choice for a first go at bow-making. There's a lot of information on this online. I've shot a few professionally-made ash bows and enjoyed it. Ash was used alongside yew in medieval Britain. Regardless of the wood type, you need a long peice with very straight grain (unless you're confident joining and laminating like most modern bowyers, in which case you need smaller pieces with very straight grain). Your best chance at getting this is with ash. Even in the medieval heydey of the famous yew longbow, we imported almost all of the staves from southern Europe, where yew trees grow faster (greater length between knots) and straighter. With what's available in your local woods, you will probably make a better bow from ash than you will from yew. It's also slightly simpler from a woodworking perspective.
What do you mean by flute exactly? I wouldn't go to much trouble with elder. It's true that it's easily hollowed, and you can make a simple whistle from it, for a little fireside amusement, using some bit of twig lying around for a fipple, or you can get a sort of panpipe thing going on. However, elder is very soft and coarse: the instrument would not be very durable, nor would it be easy to tune holes with precision.
Willow whistles (aka willow flutes) are a common woodsy trick, and more durable than elder. Such whistles can play a number of notes, by overblowing and by covering or uncovering the end. The ability to make such a whistle is more widespread than the ability to play anything resembling a tune, although some can.
If by flute you mean a tuned instrument (recorder/whistle/transverse flute), then you want something hard and fine-grained - fruit woods or beech maybe - so that (1) the hard work results in an object that will last, and (2) so that the holes can be drilled very precisely. This is challenging woodworking, but there is information online.
It occurs that a swanny whistle (willow, hazel) can play more than one note, albeit comically!
While not native, bamboo has some obvious advantages, and is widely used for instruments in the far east. Do any friends with gardens have a stand?
Bone flutes and gemshorns also spring to mind - both primitive but excellent instruments. The hollowing out is largely done for you, leaving only the tricky part - drilling and tuning.
