I think that there shouldn't be any specific mention of a time frame in V's demise nor so much prodding with a notion that there is not much time left.
Not only does it get old but it makes you think you shouldn't be doing the side quests, in a way.
For example, Vik shouldn't say you only have a few weeks left, only that you will eventually die and should generally be more hopeful of a solution or at least of V living a good life with the time he has left... This way you feel like you still have the option of investing time in the quests, especially at the start.
Of course, it should become more and more bleak but give V a bit of slack?
I think I agree. To avoid the ludonarrative dissonance problem it should have been stated that the relic will kill V, but it will likely take some time.
This is a pretty common problem in CRPGs and open world games though.
"Portals to hell have opened across the land and only you can close them!"
"Yeah, cool, I'll be right on it when I've become the arena champion and head of the thieves guild. Six months, maybe?"