Can anyone tell me whether there is a product which is suitable for taping injuries, etc which won't stick really hard to the soft skin on their feet?

I found this article for you it may help you http://www.starlingtalk.com/fractures.htm#ballwrap and this one

http://birds-of-prey.org/cm/toe-splints/, I hope it helps.

Thay are cool links Elly.

There is also a white bandage tape that is not very sticky - you can get it from the chemist.

Agree with Liv... The white tape you use to strap 'people' injuries.

For little budgie feet I would use steri strips.