Yeah, it sounds like they really haven't come up with a consistent pattern for how it all works--which is to be expected with a variety of authors and a shared world.
That said, if you want the most current interpretation, the SCAG is what we have. Here is what I can extract from it on the most current official stance:
The Place and Process
3) Souls wait on the Fugue Plane, in the City of Judgement, "often unaware they are dead."
4) The servant of a deity might come to claim a soul.
5) Otherwise, they "might wait centuries before Kelemvor judges where they go," and "decides the fate of each one."
Judgment and Consequences
6) The requirements for attaining "the afterlife they seek" is to "revere the gods according to the rites of their religion."
7) The gods judge who is worthy. If one is, "they are taken to their awaited afterlife in the deity's domain."
8) The criteria the deity uses is "Those who lived their lives most in keeping with a deity's outlook".
9) Those not in that category, who have to wait for Kelemvor to get around to judging them, include those who "have transgressed in the eyes of their favored god, or have not followed any particular ethos".
10) Kelemvor's potential judgments include the soul "serving as guides for other lost souls," and being "transformed into squirming larvae and cast into the dust".
11) Only "The truly false and faithless are mortared into the Wall of the Faithless".
Wrenches in the Plan
12) Kelemvor is viewed as "there to take each soul by the hand and lead it to the proper afterlife." This seems to imply that Kelemvor has wide lee-way. It is also unclear whether the listed punishments are intended to be comprehensive
13) The text is vague on whether or not a specific patron deity gets to do all the judging, or whether it is a bit looser.
14) "Martyrs who die that others may live are always blessed by Ilmater with a final rest and reward in the god's afterlife, should they so choose."
#14 is the real wrench in the plan. It throws a whole new light on everything and seems to demand non-traditional interpretations. Specifically, it tells us that recruiting in the Fugue Plane is allowed! The wording implies that Ilmater can and does recruit from anyone, so not only can those who were "faithless" in life get a second chance, but apparently divine poaching is allowed.
This sets a precedent for a lot of flexibility. It seems to tell us that regardless of whether or not you had a patron deity, and regardless of whether or not you lived up to their expectations other deities are allowed to recruit you while in the Fugue Plane. And since it might be centuries before Kelemvor gets around to judging you, you have a long window of opportunity.
Take that in for a moment.
To benefit from this you do have to take up some god on their offer, and they won't give you the offer unless they feel you are worthy (and by the same token, declaring a patron in life won't do you any good if you don't measure up to their standards). And there is no guarantee any god will try to recruit you at all. But Ilmater sets a precedent that if you did live according to some deity's values they might very well offer you a chance, regardless of whether you followed them (or even believed in them) in life. And in fact, given that they derive their power from having worshipers, they have every incentive to do so. This makes a heck of a lot of sense. It has been well established in lore that a soul is a hot commodity. The fiends are wheeling and dealing and raiding and fighting over them. The deities derive power from their worship. If they are allowed to go and make offers to souls that weren't all that devout in life, but they feel are close enough to their own heart, why wouldn't they? It's a straight up win for the deity. If someone didn't follow a deity in life, the biggest reason they wouldn't have any recruiting attempts made--given that most of them will at least unconsciously live according to some god's ethos with the variety of them out there--would simply be that the the deity didn't even know they were there. Perhaps the deity's servants need to be actively recruiting. Or maybe they just know when someone who lived highly in-tune with their ethos dies. Ilmater must have one or the other of those going on to be able to recruit any martyr.
Regardless, either because they reject recruiting attempts, or because no deity attempts to recruit them, some are going to end up judged by Kelemvor.
At that point, it isn't entirely clear how much lee-way Kelemvor has. According to #12, it seems like he's trying to get them to the right place. The two big possibilities that jump out to me are that either a) anyone who makes it to him gets some sort of punishment, or b) he can punish or provide whatever other consequences he sees fit. This would allow for him shucking their souls of the Forgotten Realms dead-net and letting their souls float to an afterlife that accords to their alignment.
The last option (turning them loose from the Forgotten Realms dead-net) seems unlikely, because I'm having a hard time coming up with a scenario where Kelemvor would feel that he shouldn't do that. It is however possible that he might send them to the domain of whatever deity he feels they fit the best--giving them a third chance. Perhaps the ones he keeps around as servants are apathetic, while those who get turned into larvae are just generically evil (I mean, who really wants to sift through shades of black deciding which deity to send them to? Icky), and those who go into the Wall are those who are just standing there insulting the gods and refusing to accept assignment to the realms of any of them.
Of course, the problem with that, as I brought up before, is that it just seems like that number would be so small that the Wall would blow away if a demon breathed on it.
In any event, while 5e isn't entirely clear on what happens, it does flatly contradict some previous materials, and allows for quite a variety of interpretations within the text we've been given.
(Quotes taken from: p. 20 under "The Afterlife", p. 25 under "Asmodeus", p. 31-32 under "Kelemvor", and p. 30 under "Ilmater")