There are a few options you have (I just finished a SBS quilt for my niece):
1) It sounds like it's designed for turned edge/needleturn applique. You could certainly do it needle turn & get a beautiful result. I finished 12 SBS boy/girl dolls via needleturn in 20 hours, so it's not as huge of a project as one might think.
2) You can do turned edge with freezer paper (which is removed before stitching); or you can even set your seams using mylar template plastic (make sure it's mylar, or else it will melt when ironed over). As Tartan mentioned, the parts that go under another part (for me, that was only the top of the dress going under the bonnet, part of the glove going under the sleeve, and part of the boy's overalls going under the sleeve), you want to leave that part unturned to reduce bulk.
3) You can cut everything on the inside line (with or without the extra where they overlap), window some Wonder Under (or my favorite: Misty Fuse) -- cutting away all but the outer 1/2" of the shape -- and fuse it. To make things easier on yourself, I'd make units of some of the pieces. For example, I'd fuse and stitch the hat band (if there is one) to the hat/bonnet before fusing the bonnet to the background. Same thing with the sleeve/glove on the dress & any decorative band that may be on the dress.
One other tip: I've learned to cut my background squares 1/2" larger than what the pattern calls for as some of the fabric can be taken up in the applique process. So if a square finishes at 10", I'd normally need 10.5" to do my seam allowances. With an applique block, I'd actually cut my background squares at 11" and then trim them to 10.5" when I finish the applique. Make sure to mark the center of your block & start all work in the center. It happens less with fusible than with needleturn, but I've had it happen with both methods & there's nothing more frustrating than putting all that work into a block & finding it's 1/8 - 1/4" too small.