Steamed it

First I steamed the wobbly limb tip to an even reflex. I left the bend at the handle as it adds some character and a bit of deflex in that limb with it's twisted grain will help keep the stress down.
That was pretty straight forward using my usual steaming technique which has been pictured before.
Steaming the handle was a trickier job as I only wanted the handle region getting hot and bending.
I narrowed the handle and shaped it slightly to encourage the bend to occur where I wanted it. To get the steam in the right place I used an old 2.5L plastic container (it had a bit of windscreen washer fluid left in it, so I topped up my washer bottle first). The steam hose from the wallpaper stripper was pushed into a hole cut in the cap and I cut slots/flaps in either end to push the bow through.
The first attempt didn't quite straighten it enough so I re-did it, this time with the bow clamped in position onto a piece of 3x2 with my plasic steam chest around it. The advantage of this being that there was no cooling time between getting the bow from the steamer to the clamped position. The taper of the limbs is clamped flat against the 3x2, this over-bends it slightly, but there is some spring back after it's cooled, it's just trial and error to gauge how much to bend it.

The second arrangement is shown right, I hope you can see what's what. It shows that you don't need fancy equipment to do steam bending, a wallpaper steamer and a few good clamps is about it, some rag to help keep the steam in helps.

A word of caution, once the steam starts keep your hands well clear, it's nasty stuff steam.

The results are pretty good. During the first steaming the area with the twisted grain took a bit of a twist. I put a G clamp on the limb while it was still hot and applied some force to twist it back as it cooled.

The end result is going to be a bit like an American Flat Bow (AFB, which they sometimes call a 'Longbow' ), I'll probably taper the limbs to a very narrow tip a bit like a pyramid bow (thats a bow of near constant limb thickness which is wide near the grip, tapering to a narrow tip E.G nearly all the taper is in the width) This bow isn't wide enough for a full pyramid style so it will be a bit of a hybrid...well it's a stave bow and it will turn out how it turns out, we don't actually need to label everything.
It sometimes amuses me when people refer to specific prehistorc bow remains as if all the bows at that time would have been identical. A bow is whatever you can make out of the wood, it's your challenge to make it as good as possible, or maybe as quick as possible depending on your circumstances. Whoops I'm getting preachy.