Part One of Two:
I finally thought of a movie and tv series which might possibly - repeat possibly - be the record holders in characters dying over the longest screen time for their respective forms.
I have never seen the French film L'Abre de Noel/The Christmas Tree (1969) (110 minutes) but I have heard about it. I think it is quite possible that a character in it is dying for longer than the 91 minutes for a character in Dead Man (1996). Warning; it has been called "the tearies film of the 60s".
I also don't remember seeing Run For Your Life (1965-1968) in which the title charcter Paul Bryan learns he has only a short time to live and so travels the world trying to live as much as he can. So I don't know how often his medical condition is mentioned in the various episodes, though the opening credits of every episode explained his situaiton.
As far as I know Paul Bryan survived the 48 minute April 15, 1965 episode of Kraft Supsense Theatre and all 86 episodes (each about 45 to 48 minutes long) of Run For Your Life (1965-1968) or about 3,918 to 4,176 minutes.
I don't know if the dying character has to actually die onscreen before the end of the moive or television series for it to count, or if it alright if they survive to the end if they are clearly dying.
And I can't help thinking that maybe there are other movies and tv shows with similar plots.
The rest of this answer is some some real examples of people and animals living for a long time while dying.
Part Two:
In real life it can be very hard to tell what killed someone, thus it can be hard to tell how long ago the fatal injury or other cause of death was inflicted.
On 27 May 1942, assassins wounded Reinhard Heydrich who died on 4 June, 8 days later. Though he was fatally wounded, Heydrich ordered his driver Klein to chase the assassins,and was still conscious until June 3. Thus it is possible to kill someone, or at least fatally wound them, and for them to still be conscious and capable of taking actions and giving orders for days. The assassins wer tracked down and killed on 18 June.
In the Second Battle of Fort Fisher one of the Union brigades was led by 20-year-old Colonel Uriah Galusha Pennypacker, who carried a flag to the ramparts of the fort and was shot down. Pennypacker was not dead (as was believed at first and was discharged after ten months in the hosbital. Pennypacker eventually died on October 1, 1916, and his dotor said it was from his old wound. If the doctor was correct, it took 51 years for the wound to kill Pennypacker. Pennypacker remained in the army until 1883, and may have sometimes given orders which resulted in the deaths of persons, which would make him a pretty dangerous "dead" person.
While lying "dead" on the ramparts of the fort Pennypacker was actually imobile but conscious,and in alter interveiw he said his men dememand a blanket to cover his body fromt he rebel soldier whohad shot him. When the rebel refused, Pennypacker's men kill him. So that is an example of someone ftally wounding some and yet dying 51 years before their victim.
So if anyone every made a biographical tv series about Pennypacker a very long part of it could be considered to be his dying scene.
And since I don't know about all the tv series ever made, I can't guess how long some might have streatched out the death of a character.