What can be the cause of bleeding gum and nose bleeds and watery eyes?

See your doctor. The watery eyes may or may not be associated with the bleeding, or may be due to congestion from the nose bleed. Recurrent nose bleeds and bleeding gums should be evaluated by your internist. Do not delay visiting him/her.
Dryness. The most common cause of watery eyes is dry eye syndrome. Dryness in the environment can also cause a bleeding nose, however bleeding gums is not usually related. I would try to make an appointment with your primary doctor to further evaluate your issues.