According to Theravada tradition, was Siddharta Gautama illuminated after meditating under the Bodhi tree?

According to Theravada tradition, was Siddharta Gautama illuminated after meditating under the Bodhi tree?

There's a summary here:

The Buddha and His Dhamma

Now he was alone, and complete solitude allowed him to pursue his quest undisturbed. One day, when his physical strength had returned, he approached a lovely spot in Uruvela by the bank of the Nerañjara River. Here he prepared a seat of straw beneath an asvattha tree (later called the Bodhi Tree) and sat down cross-legged, making a firm resolution that he would never rise up from that seat until he had won his goal. As night descended he entered into deeper and deeper stages of meditation until his mind was perfectly calm and composed. Then, the records tell us, in the first watch of the night he directed his concentrated mind to the recollection of his previous lives. Gradually there unfolded before his inner vision his experiences in many past births, even during many cosmic aeons; in the middle watch of the night he developed the "divine eye" by which he could see beings passing away and taking rebirth in accordance with their karma, their deeds; and in the last watch of the night he penetrated the deepest truths of existence, the most basic laws of reality, and thereby removed from his mind the subtlest veils of ignorance. When dawn broke, the figure sitting beneath the tree was no longer a Bodhisatta, a seeker of enlightenment, but a Buddha, a Perfectly Enlightened One, one who had attained the Deathless in this very life itself.

For several weeks the newly awakened Buddha remained in the vicinity of the Bodhi Tree ...

I think that's a paraphrase of what's said in several suttas, e.g. MN 26 and Ud 1.3, and others.

But Theravada tradition says that one cannot reach Nirvana while the five aggregates exist, since them are dukkha.

I don't think so.

The first noble truth (in SN 56.11) says, something like, that that the "clinging aggregates" or "grasping aggregates" are suffering (or if not "are suffering" then something like, "are unsatisfactory") ...

Rebirth is suffering; old age is suffering; illness is suffering; death is suffering; association with the disliked is suffering; separation from the liked is suffering; not getting what you wish for is suffering. In brief, the five grasping aggregates are suffering.

jātipi dukkhā, jarāpi dukkhā, byādhipi dukkho, maraṇampi dukkhaṃ, appiyehi sampayogo dukkho, piyehi vippayogo dukkho, yampicchaṃ na labhati tampi dukkhaṃ—saṃkhittena pañcupādānakkhandhā dukkhā.

... that dukkha arises with craving ...

It’s the craving that leads to future rebirth, mixed up with relishing and greed, taking pleasure in various different realms. That is,

yāyaṃ taṇhā ponobbhavikā nandirāgasahagatā tatratatrābhinandinī, seyyathidaṃ—

craving for sensual pleasures, craving to continue existence, and craving to end existence.

kāmataṇhā, bhavataṇhā, vibhavataṇhā.

... and ceases with the cessation of craving (it doesn't say "with the cessation of the aggregates") ...

Now this is the noble truth of the cessation of suffering.

Idaṃ kho pana, bhikkhave, dukkhanirodhaṃ ariyasaccaṃ—

It’s the fading away and cessation of that very same craving with nothing left over; giving it away, letting it go, releasing it, and not clinging to it.

yo tassāyeva taṇhāya asesavirāganirodho cāgo paṭinissaggo mutti anālayo.

I think that "with nothing left over", there, doesn't mean "without aggregates", it means with no craving nor any tendencies that gives rise to craving, e.g. āsava and anusaya.

Conversely, the Buddha is liberated -- I think that's said or implied in so many suttas that it's difficult to choose one reference to cite, but see e.g. Why is the Buddha described as trackless?

Or how otherwise do Theravada explain that Siddharta Gautama did not cease to exist after the meditation under the Bodhi tree?

I think that you're associating the Buddha with the aggregates, i.e. that you're arguing something like:

• Siddharta Gautama is the aggregates
• The aggregates didn't cease to exist
• Therefore Siddharta Gautama didn't cease to exist

However identifying the aggregates as "self" is contrary to the anatta doctrine -- and the Anuradha Sutta for example says it's inappropriate to regard the Tathagata as being the aggregates, as being in the aggregates, as being without the aggregates.

I think that Theravada (and/or the Pali suttas) identify two types of nibbana, i.e. nibbana-with-remainder (with aggregates remaining) and nibbana-without-remainder -- the latter is called parinibbana, see e.g. the "parinibbana sutta" -- see also e.g. AN 4.23 which refers to anupādisesa:

From the night when a Realized One understands the supreme perfect awakening until the night he becomes fully extinguished—through the natural principle of extinguishment, without anything left over—everything he speaks, says, and expresses is real, not otherwise.

Yañca, bhikkhave, rattiṃ tathāgato anuttaraṃ sammāsambodhiṃ abhisambujjhati yañca rattiṃ anupādisesāya nibbānadhātuyā parinibbāyati, yaṃ etasmiṃ antare bhāsati lapati niddisati sabbaṃ taṃ tatheva hoti, no aññathā.