You are lucky, your gun is an ideal candidate for an invisible repair.
Most PVA adhesives will create a bond that is stronger than the wood itself, but no doubt someone may also suggest alternatives that may be even better.
The "hot tip" is - pre-clamp (practice) as clamping a gun is always going to be awkward. Only when you have rehearsed the drill should you "go for glue".
No need for any extraordinarly pressure (which may distort) - as long as the gap is closing perfectly then you are good to go. I could even imagine the gun perhaps inverted in a vice retained by the butt cheekpiece area and the very weight of it closing the gap, or perhaps supported at muzzle and buttpad (right way up). You will have to find what works for the damage you have.
Not to worry about injecting the glue, some have useful applicator nozzles (evo-stik wood glue) but basically apply the pressure with the nozzle to the crack, and then "work" the crack open and closed so the glue will creep. Do it a few times and the glue will have gone where it should.
Once clamped or otherwise, wipe off the excess glue that is squeezed out and let it be for at least a day!
**WANTED**: WEBLEY PATRIOT MUZZLE END; Any Diana/Original mod.50 parts, especially OPEN SIGHTS