Which patient should be treated first the medical asthmatic patient presenting with apnoea or a trauma patient with severe bleeding on the proximal right extremity?

It's a matter of priority, what is more important loss of blood or lack of Oxygen. If the trauma patient is conscious put a bandage in their hand and tell them to apply pressure and elevate the limb if there is no sign of a fracture. Then you can concentrate on the medical patient. Put the asthma patient into the semi-seated position get them to take their inhaler and don't forget to call for the EMS.