You also might want to consider how dry the air in your house is. I know in the winter when we try to keep the humity levels low in the house my sinuses dry out and I will wake up many mornings either getting a massive nosebleed when I sit up or with blood all over my pillow. you may just need more moisture in your house or bedroom if that is where you seem to get the most nosebleeds. As for the headaches, another thing to check is your quality of sleep. I have insomnia and when I get little to no sleep I notice a huge increase in the headaches. If your blood pressure is normal along with other issues in the comment you might want to consider asking your Dr. about poss. getting a sleep study done. This would tell you if there is any thing abnormal such as not breathing, waking up not enough rem sleep ect that can be helped and poss. get the headaches under control. Good luck!!!! I'm still trying to find out what is causing my headaches.... I am begining to think it is the Dr's I am dealing with!!!!!
Obvious first stop is to get your blood pressure checked. Hypertension can cause headaches, and sometimes the nose will bleed to attempt to lower the pressure. Morning headaches can also indicate pressure in the skull, rarely. Definitely worth going to your doctor and having your blood pressure taken, and maybe a simple neurological exam to rule out intercranial pressure. Less dramatically, morning headaches can just be from dehydration from maybe sleeping with an open mouth or being too hot, and bleeding noses can be from eg polyps in the nasal passages. Could be the two are not related. Hope I've given you some thing to think about. Have a headache myself right now, but I know the reason for mine.