My first instinct would be to use the name the character was introduced at. So in your case, in the narrative it would be Albert and not Al.
Technically, "Albert" was never introduced. He begins the story as "Al" with no explanation, while his sister is introduced as "Betty", but whom he calls "Be" in his dialogues as a pet name.
I know it's appropriate to have those around them refer to each other by the newer names, but you rarely specify names in dialogue, so the name changes become largely invisible. I was hoping to make them a little more obvious, at least for a chapter or two, by emphasizing how their perceptions of who they are has changed (their becoming their own people, rather than who they're expected to be).
I might be expecting names to do more than they were intended to do, though.
One thing I typically do, especially when introducing a character after they get independently mobile and have a nickname is to include the nickname in italics when I introduce them, then will often use that to refer to them.
Ernest, again, these weren't the names they were introduced with (aside from Al), they're 'adopted names' that the individual characters take on to set themselves apart from their previous identities (to emphasize they no longer feel they belong to their previous lives). It's a different mindset I'm trying to capture, via the use of names (in the narrative, since the use in dialogue is rare).
I guess I can have them 'introduce' themselves to new people, where I use your italic technique', to help cement the name, but those situations are several chapters removed, and may occur a bit late to signify the change which has taken place (and potentially confusing readers even more).
I think Ernest is right that you have to be very clear when a name gets changed.
OK, I think I'm doing that, as there's a transitional scene, where the foursome verbally agree to change their names (at least amongst each other). I then began to reference the new nicknames in the narrative to reflect their new reality, but was afraid I might be making a mountain out of a molehill.
By the way, it's a tale of alienation (by society) and isolation (by the group). The name changes represent their setting themselves apart from the wider society. It's more of a mental image thing by the group, as they change their membership in society.