What's the best way to repair or replace window sill & apron that's been sawed off?
Drop the wood putty idea. That is just not workable for this situation. I believe you are on the right track with the apron...just replace it...but fix the sill as below before installing the new apron. 

For the sill I would recommend getting some paint stripper and taking off the existing paint on the sill. Once you have exposed the cut off area of the sill you can evaluate whether the cut is nice and straight and smooth. If not it may be necessary to make it so. The next step involves getting a piece of nice wood of the same thickness (pine would be good to work with). Cut out a triangle of the wood that will be glued to the cut off area. When you size and cut this piece leave it a bit larger in the direction facing into the room so that once attached you can plane and and sand the front edge to match the existing sill. 

For attaching the triangle of wood to the cut corner of the sill you will be using a good quality carpenters wood glue. Success with this is going to depend on several things:


How nice of a match you can get between the edges of the mating parts. 
Ability to clamp the triangle piece into place.
Consider the possibility to screw or nail the triangle piece into place although this could be very difficult due to the approximate 45 degree angles involved.
Consider the possibility to use dowels or biscuits to hold strengthen the joint when it is glued. Using these does require special tools and may be tricky to use when dealing with the still installed existing sill board.


Once you have success in attaching the new piece you can plane and sand the the corner till the surface and edge merge points are smooth and even. After that you can prime and repaint the sill area. (Remember the stripping recommendation at the beginning...you will appreciate that when it comes to repainting and how you can get it looking nice. It would be a pain to try to make the corner look good if you just tried to patch paint the corner).