Please beware, major spoilers below.

In the final scene, if you had previously used the bronze sphere and discovered your true name then certain dialogue choices will lead you to an option to 'unmake' your mortality, rather than destroying or merging with it. This ending leads to a short cut-scene of your mortality being unmade followed by the credits.

In more detail:

Your mortality, The Transcendent One (TTO) mentions how he doesn't wish to kill The Nameless One (TNO) but simply make him forget so that he can live in his fortress in peace. TNO states that he can now remember everything and death no longer makes him forget. This means that there is nothing that TTO can do to stop him returning to this fortress time and time again. During this conversation, TTO claims that both he and TNO are linked somehow, but TNO counters this by asking him what his true name is, as surely he must now know if they are linked. TTO doesn't know his true name, which suggests that they are not linked. The Nameless One then uses his knowledge of The Transcendent One's true name to unmake him.

The ending and dialogue suggest that only The Transcendent One is unmade not The Nameless One. The dialogue states that TNO is unmaking TTO, not them both.

Does this ending mean that TNO remains immortal and has escaped his punishment of an eternity in the blood war by 'unmaking' his mortality? Has The Nameless One become truly immortal?

Great question, I was wondering about that as well. I think you are indeed correct.
When we chose to unmake TTO, we see him destroyed in the ending cinematic, but we don't see the second cinematic where The Nameless One dies and enters the Abyss as his punishment.

It would seem that in destroying his mortality, the Nameless One has achieved true, eternal immortality. Thus, he has tricked fate and escaped his punishment.

I always thought the opposite. That by unmaking his mortality TNO also unmade himself. And that is why we do not see what happened to him after the first cutscene is over.

But the beauty of this game is that it is opened to many interpretations. :)

Edit: Oops, I've just realized that this was a necroed thread...

No problem friend, this was pretty much the only discussion thread on the internet that I could find about this specific ending, so I decided to revive it.

Thank you for your intepretation, one could view it this way as well. Good insight.