I don't think such a society could develope on it's own.
The selfish human desire is both nature and nurture. It is built into us (selfishness means more food means better chance to reproduce; therefore evolution makes us selfish), and built into our society, too (ex. capitalism). The urge to own private property is an example of selfishness. But selflessness is, too, built into us, though not to the extent that selfishness is (selflessness means community means protection; therefore evolution creates a selfless element within us. This is furthered within our cultures regarding respect, kindness, honour, and other traits that integrate one into a society, and make us protected). To be able to give up on property is to be selfless.
I do believe that, with enough nurturing, the human selflessness can be made to overcome the human selfishness. This, however, takes dedication, effort from others, and ideally, a lack of preexisting selfish notions in the mind. And to instill such selflessness in an entire people, this would logically have to be pushed to the extreme. As such, the development of this collective society would take extreme dedication and effort from an outside, nurturing source, be that divine, or a pregenitor or "parent" civilisation; this is impossible in the scenario you describe, as there is no nurturing force (other than perhaps a divine one, though I don't believe in such a thing).
Even if we are to assume there is a divine nurturing force, we would still be missing the other component in the development of a selfless society: the lack of preexisting selfish notions. In the societal breakdown you have described, the human selfish instinct overpowers the selfless one (after all, if one find himself in a wasteland, he would naturally prioritise himself and his family). Therefore, these selfish notions are ingrained in the people of this cataclysmic world, and it would be nigh impossible to nurture it out of them.
As such, because of the lack of a nurturing force, and the preexisting selfish values of these post apocalyptic people, such a society is impossible to form.
Alright. Then let us imagine a scenario where this selflessness is possible: this world must be bountiful and abundant with resources, with no dangers as to not provoke the natural selfishness in people. The people of this world, as such, are mellow and easygoing, not very hardworkers (as they have no reason to work very hard when all is abundant), and not desiring much (as there is nothing left to desire).
In this world, there must also be a nurturing force to mentor these people, and bring out the selflessness within them. As humans are inclined to do as their mentors do, this force must necessarily be selfless, else the people would copy the mentor's selfishness. I see two possibilites for this mentor:
The first is that the mentor is a group, perhaps a progenitor or parent civilisation as I have previously described. I will presume this civilisation is comprised of a species bound by evolution (I will cover the opposite later on), and as such, has an inherent selfish component. This raises the question of, who mentored them to be selfless? And who mentored the mentor of the mentors? And so on. The logical endpoint: there must be an inherently selfless mentor, with whom this cycle breaks.
The other option is to simply skip this cycle, and assume the mentor is inherently selfless. As all species subject to evolution are inherently selfish, this mentor must necessarily come about outside of evolution, and is therefore what we may label "divine." (This is not to say that this divine being has those powers we so often associate with divinity, however.) I will refer to this being as Mentor. As Mentor is completely selfless, even if they were a collective, the goal would be the exact same: the best for everybody around them. Therefore, as the objective is the same, we may refer to Mentor as a singular entity. As for Mentor's role in this selfless society, I see two options:
One: Mentor claims no power over the people, and takes on a soley guiding role. In a selfless collective, this leaves a gigantic vacuum of power. As there are no outside dangers, this vacuum can only be filled by one of the people. As these people are taught selflessness, and their human selfishness is simply dormant, it is logical that eventually, one (or multiple) would awaken their selfishness, and come to fill this power vacuum---and these mellow, non-hardworkers would not be able to fight back. These new, selfish rulers would undoubtedly shape this society to their needs, and ultimately, corrupt its selflessness (see: the Soviet Union).
Two: Mentor fills the power vacuum. This is the only scenario in which, I believe, such a selflessness can be taught as to allow for the giving up of private property and the creation of a collective.
To conclude: it is my belief that a collective as you describe, with a complete lack of ownership, is possible only under the best of conditions, with complete abundance and safety, and requires direct divine rule. That is to say, it will never happen.