Which glue works best for paper sculptures
Given your criteria the glue I would think should be tried to see if it fits your needs is cyanoacrylate glue, or CA glue. This can be commonly refereed by brand names like Super glue or Krazy glue as well. They are advertised and being very versatile when it comes to applicable mediums and have very fast drying times. 

Looking at one brands technical specifications as a base line we can see that it


  Dries in 10-45 seconds, no clamping required.


Most CA glues dry at or near those times as well. Careful working with the stuff as you a likely to get some on your skin and it bonds skin very easily (Recently used it to repair my glasses).  

Now what you could so with this is use it as a placement glue. Using a little just to get the leaf in place so that you can continue with a longer setting glue without concern of the leaf moving. 

Would clamping options apply here?

I don't know how your leaves are made nor how they touch the tree but have you tried clamping options instead of looking for a fast setting glue? Some simple like paper clips( which you can mold to conform to shape if need be) or even something like binder clips? Perhaps if you have something small and light weight to help hold the leaf in place your choice of adhesive might no longer be a concern.