Choose a glue that has a long shelf life. eg an epoxy or titebond 3. Good epoxy can be expensive but lasts a hell of a long time for a modern glue, so its money well spent.
A urea glue is probably the best for wood to wood gluing, and bowmaking, but has a short shelf life. These days its hard to buy in small amounts. Where I live, the smallest amount I can buy is 5 litres. That's an industrial quantity, which I couldn't get through with bows only before it expires. Yes it is also very expensive, and then you have to add in extra dangerous goods shipping, because its made to order, not retail.
The mix with water(not resin), powder urea type has a longer shelf life if stored properly and can be bought in small amounts, and is relatively cheap. Frequent complaints of out of shelf life products being sold, so do a test before committing to gluing up bow quality material.
You originally mentioned sinew horn bows. You only need animal glue for that. It lasts indefinitely if stored in dry/powdered form.
Isinglas is often used which is expensive, but you don't need much.