Don't use birch, it's useless unless you know exactly what the density is and has super fine grain. Ash is a fantastic bow wood, but again it requires a very good piece to make a good bow. You'll find that using any old piece of English ash and making a basic bow out of it will give a fairly good bow to start with (provided it's tillered very well with no hinges) but will quickly take a lot of set as ash is prone to it.
That being said (and I'm not trying to be negative!) give it a go and you'll probably love the process far more than the outcome, and you'll just keep makin' em. There's a lot of skills required before good bows start coming such as careful reduction of staves, feeling taper and widths rather than relying on dimensions and numbers, tillering of course is a whole load of learning just on its own, finishing, string making, bracing and so on. It's fantastic fun though, and highly addictive so I say get out there, chop a bit down and see what happens!
In short - ash is brilliant. I bet you've got yew somewhere though, and hazel. Del The Cat on here is a big fan of hazel and will have lots of advice, but I've not used it yet so couldn't offer advice. I've used a lot of yew though, and it's very user-friendly - you can abuse it to hell and it will still make a decent bow. You don't need much of it to make a quick survival bow.