To answer this we need to know that that there are always doctrinal parallels among religions despite differences. But this is different from the question of which religion we should ultimately follow. Islam is certainly the final Divine religion with a Holy Scripture that has survived history intact with a very rich doctrine and relevance to today and with a promised future, hence why it is the right religion to follow.
Comparing Islam with Buddhism though is tricky. Buddhism has an ambiguous origin and some ambiguous doctrines but also with some interesting parallels with Islam. It is either that Budhha's doctrine was just a product of his mind in which case it wouldn't be much interesting or that it was result of his actual encounter with some spiritual truths. It is mostly the latter case that can make him interesting I believe because clarifying that position of Islam as a revealed religion about another revealed doctrine is meaningful.
Below are some notes on some apparent parallels between Islam and Buddhism with any eye to the question.
There is indeed an opinion among some Muslim scholars (though not mainstream) that Buddhism in its original form was probably a Divine religion. The idea of "karma" in some interpretations indicate birth into a higher life, not coming back to the natural universe which is not accepted by Islam if it denotes a universal process that affects all men. In the former case, it basically parallels Islamic eschatology in which man dies and enters Barzakh with residence in barazah differing in actual fact, quality and length for different individuals, with Divine justice established upon Qiyama after a very long time. According to some accounts there are many many stages to Barzakh and Paradise with each progression involving some sort of death and rebirth into a higher level. This is broadly consistent with Buddhist doctrine of re-birth into other realms.
The idea of liberating from the cycles of birth and rebirth then would parallel the Islamic doctrine that Prophets and very pious believers don't actually go to barzakh or paradise or face the day of Judgment after death but they will immediately go to the vicinity of Allah without the toil of agony, barzakh and judgement. In this case, pleasures of paradise for them would be insignificant as they desire only Allah.
The ethical doctrines of Buddhism have quite concrete parallels to that of Islam. No harming others, no lying, cheating, greed, arrogance, sexual misconduct and so forth.
Merit transfer has particularly interested me for its parallel with the doctrine of shifa'a in Islam. A pious Muslim can intercede for his family members and associates, and each Prophet interceding for his own ummah.
And finally as for your particular question about Nirvana and its stages there are apparent parallels too.
Eliminating emotions and stimuli resembles the Muslim doctrine of zuhd or abstinence which involves moderating worldly desires and only sufficing to basic life necessities. Not reacting to external stimuli may parallel the doctrine of riza and tasleem (staying content and surrendering to the Allah's will in the face of worldly hardships) or overcoming wasawis (temptations).
The idea of non-self may effectively parallel the concept of khushu which means humble submission in Islam. A state that is recommended specially during salat. Or the Islamic doctrine that we are all created from one self according to Quran hence our individual selves is denied or downplayed.
As for the idea of Nirvana, it is a more tricky concept. Scholars have different interpretation of what the state actually is. Is it just the state of not identifying with anything? i.e. just a negative state? Then it is difficult to see what would be the point (use, benefit) of such a state. Or does the negative description have a positive side to it, like identifying instead with a higher being, god -- Brahman probably? The idea of a supreme being or Brahman is believed to have been rejected by Buddha but then who really knows? Some buddhist schools do actually incorporate the idea of a supreme being or god. But because Nirvana is associated with liberation from life cycles it can't be only a negative state. And the characteristics of nirvana as a state are very consistent with those of Brahman as a deity in Hindhuism.
In the light of above then existence of certain Islamic theological interpretations that understand Allah as a formless, immutable, absolute and pure being above all categories which as a result is not subject to any change, affection, division or any particularizing property would be an interesting parallel to Brahman and Nirvana.
Also the idea of a mental state when you don't actually identify with any particular thing and become fully content with anything in life, then may be interpreted as having reached the state of fana and the subsequent vicinity with Allah according to this Islamic theological account where you no longer identify with worldly and even next-worldly attractions and forms and realize that the only real and substantial reality and goodness is Allah which is above all creatures.
Then we can answer the question whether a person like this would be affected by heaven and hell. Assuming that the state really represents full vicinity of Allah, it would be a state above and beyond heaven and hell. According to this theological view, saints who have reached this state have experienced many spiritual hardships and joys that other people would normally experience in barzakh, janna and jahannam. So they have already transcended these higher realms in their lifetime and won't experience them after death. Of course, this assumes that they won't have any unaccounted bad deeds (or any) after death that require them to face Divine judgement or punishment. Therefore they won't be affected by pleasures and pains of paradise and hell but not because they are somehow immune to Divine compensations but because they have already settled and transcended them in their worldly lifetime.
But if Nirvana is not such a state i.e. one with real metaphysical substance to it, it would be certainly overcome by tough encounters after death because one who has developed some insensitivity to worldly pain and pleasure can't be said to be able to sustain it when facing the far more overwhelming experiences of life after death. Trials of afterlife are incomparable to worldly trails.
These were just random thoughts from my mind that were to address the OP's contention of similarities with Islamic doctrine.