"Urgency" comes close to what I want. For example, a painful sensation has urgency (may be more or less urgent), and may compel me sooner or later to change position, take aspirin, etc. Similarly, a full bladder has a quality that compels an obvious response.
What I don't like about "urgency" is the emphasis on how soon the response needs to come.
I like "exigent", except, as with "urgent", there seems to be emphasis on timeliness. Also, there doesn't seem to be a word for how exigent something might be ("exigency" means that which is required not how much something is required). Another problem is that "exigent" also means "requiring or calling for much" which is not what I want.
"Demanding" is good, but "demandingness"??
"Pressing" (and ?"pressingness") and "imperative" (?"imperativeness) are also near what I want. "Priority" too is close, but I think is better used to describe a property of tasks rather than perceptions or thoughts. Also, "priority" emphasizes the ordering of competing options, which is not what I want.
I would like to say something like, "The thought of one's own death has great urgency, even though there is nothing to be done about it," but I don't like "urgency" here because I don't want to highlight how soon it needs to be done, rather, how much something needs to be done.
Edit: To clarify in response to comments below, I am mainly interested in a noun that names a property or quality of some mental phenomenon (thoughts, sensations, emotions). Adjectives would then be applied to the property.
To give a concrete example, 'height' (a noun) is a property of a person. A person's height could be tall or short (adjectives). Likewise, an emotion might have urgency. Its urgency could be urgent or not urgent.