Despite the contradictions in the Mahābhārata (even in BORI's Critical Edition), the most likely thing that happened was that Pāṇḍu was cremated in the forest immediately after his death. Mādrī then performed satī and died along with him (sahagamana). The ṛṣis then carried their dead remains (not the bodies) back to the city. And it is unlikely the funeral rites were performed again by Bhīṣma, Dhṛtarāṣṭra and others.
Ādi Parva (Sambhava Parva)
I am still embracing my husband and have not let him escape. Nor has my desire yet been satisfied. O elder one! Please give me permission. The best of the Bharata lineage came to me for the sake of desire. With his desire unsatisfied, how can I let him go to Yama’s abode? O revered one! If I live, it is certain that I will not be able to treat your children and mine in the same way and that sin will touch me. O Kunti! But you will be able to bring up my children as your own. The king has gone to the land of the dead because of his desire for me. Therefore, my body should be burnt with the king’s body. O revered one! Do what is pleasurable to me and burn them together. Watch over the children and think kindly of me. I cannot think of anything else I need to say.
Having said this, the illustrious daughter of the king of Madra, wife by law to that bull among men, climbed onto the fire of that funeral pyre.
Vaishampayana said, ‘The maharshis who were the equals of the gods performed Pandu’s last rites. Then the ascetics assembled and consulted each other. “For the welfare of the kingdom and the country, the great-souled great ascetic came here, to live among ascetics and practise austerities. King Pandu has gone to heaven and has left his sons, who have just been born, and his wife as a treasure we must look after.” Having consulted with each other, those who are engaged in the welfare of all beings decided to go to the city of Nagasahrya (Hastinapura), with Pandu’s sons in front of them. Those generous and accomplished ones decided on the journey, so as to give the Pandavas to Bhishma and Dhritarashtra. As soon as they decided, the ascetics set out at that very moment, with Pandu’s sons and wife and the two bodies. 312
312 There were parts of the bodies that had not been burnt. Having said this, there is some contradiction about when the bodies were burnt.
Debroy, Bibek. The Mahabharata: Volume 1 (pp. 315-318).
Later in Ch. 117, one of the sages instructs Dhṛtarāṣṭra to perform the funeral rites again:
Now perform those rites that should be performed for them. These are the two bodies and here are the supreme sons [Pāṇḍavas]. Let these scorchers of enemies and their mother be respectfully received with rites of welcome. After the performance of the funeral rites, let Pandu, the upholder of the Kuru lineage, extremely famous and knowledgeable in all aspects of dharma, gain the right to ancestral offerings.
The above is most likely an interpolation (late addition).
Another possibility is that the above account i.e., both dead bodies being brought to and the funeral rites performed in Hastināpura, is true, and some later editor or scribe who wanted to glorify the satī practice made Mādrī (who probably died along with Pāṇḍu due to some communicable disease) commit satī and introduced the satī episode in between Chapters 116 & 117.
This unresolved contradiction is also listed in Interpolations In The Mahābhārata by M. A. Mehendale of BORI:
Another similar example is from the Ādiparvan. It is usually believed
that Mādrī committed satī after the death of Pāṇḍu. In support of this belief
we have a specific statement in the Mbh. which occurs without exception
in all the versions and hence forms part of the constituted text. That statement
tells us very clearly that Mādrī mounted the funeral pyre of Pāṇḍu and burnt
herself with him. (taṃ citāgatam ājñāya vaiśvānaramukhe hutam / praviṣṭā pāvakaṃ mādrī hitvā jīvitam ātmanaḥ 1.117.28; also 1.116.31). However, it is not so well known that immediately after this stanza we have another one which also occurs in all the versions and hence forms part of the constituted text.
The latter stanza directly contradicts what is said in the preceding stanza.
According to this stanza the sages who, after the death of Pāṇḍu, brought
Kuntī and the young Pāṇḍavas to Hāstinapura, also brought with them the two
dead bodies of Pāṇḍu and Mādrī. They handed them over to the elders of the
Kauravas and asked them to perform the due funeral rites (ime tayoḥ śarīre dve... pretakārye ca nirvṛtte pitṛmedhaṃ mahāyaśāḥ/ labhatāṃ... pāṇḍuḥ.../
/ 1.117.30-31). Then, in the following adhyāya (1.118) we have a detailed
description of the cremation of Pāṇḍu and Mādrī. Both these accounts could
not be true at the same time. Either Pāṇḍu was cremated on the Śataśṛṅga
mountain and Mādrī mounted the funeral pyre, or cremation did not take place
and the dead bodies were brought to Hāstinapura. Nīlakaṇṭha's feeble attempt
to remove the contradiction by interpreting śarīra 'body' as asthi 'bone' is not
convincing. Sukthankar, following Hopkins, has already expressed his
dissatisfaction with Nīlakaṇṭha's explanation.
Clearly here, as in the above case, we have conflation of two different accounts of the same incident. The interpolator while borrowing stanzas from a different account and inserting them in his own has shown scant regard for the contradiction his interference with the received text involved.