I'm hesitant to suggest this as the OP states only hand tools are available, but an investment in a router plus a bit or two if funds are available could be a solution that reduces the amount of handwork, and probably improve consistency in the curves produced. A router could be employed to create both the joints between the vertical supports and the horizontal member, e.g., a tongue and housing or similar, and the concavity in the faces of the vertical parts.
To create the concavity, at least in basic profile, in the verticals would need a further investment in materials, e.g., softwood plus board material, plus some hardware, e.g., washers, nuts and bolts, screws, that sort of thing. These materials could be used to create a pendulum mechanism to which the router is attached allowing it to swing in an arc. Of course, this is just the kernel of a method and some ingenuity would be required by the OP to make the rig so that the radius of the swing is adjustable and the means for holding the parts firmly in place whilst being shaped. There would also be a need to create lateral movement of some sort, of either the router or the part being shaped, between swings of the router.
Going with this method is sure to be reasonably technically challenging, fairly slow, tedious, dusty and noisy, but it's not new having been done before. On the other hand, doing the job with just hand tools is also likely to be relatively slow and tedious requiring some quite good technical skills.
All the above just another option as an alternative to the obvious and simple (sic) solution of a bit of digital drafting and chucking the parts onto your basic five axis CNC machine along with its requisite dust extraction system, ha, ha. Slainte.
