My teacher has pointed out she can hear my nails clacking on the keys - Yet I've cut them as short as physically possible - there's nothing left to cut - any further and they'll be bleeding. Is there any way to get the nail bed to move further back up my finger?

(I noticed in a classical guitar forum people were asking the opposite question)

How odd. Only thing you could do is keep biting your nails back til they hurt - that'll eventually push the nail bed back if you keep biting. But I really wouldn't do that. If you have cut back so there is practically no white showing and you can still hear them clicking, well, I suppose there are a lot of poeple who are the same! Is it really that noticeable?

Most likely your finger tip is TOO vertical. You could try this: Put your hand in its usual position on the keyboard and notice if your nails are touching the keys. Now gradually lengthen your fingers, one at a time, by allowing the first and second joints to open until the nail is no longer touching the key, but instead you are pressing it with the fleshy pad of the finger tip, just beyond the nail. This is your correct basic finger position.

As MT points out, you should be playing with the fleshy bit of your finger, not the tip. Playing with the finger tip is unnatural.