My last comment was apparently hidden without my knowing. I want to give information from a historical perspective on the subject of the "eternal, uncreated" nature of the Quran since it informs why the surah aren't represented in chronological order. I'm currently agnostic on the issue since my own long-term research into the topic has made me suspend my beliefs long ago. Now I'm more interested in the historical information, which is what I want to provide here. So let me try this again:

According to the Quran itself there is a primordial "Heavenly Quran" that is as eternal as Allah, not created or written by Him. Most Muslims are taught that the Quran has been perfectly preserved and is synonymous with the Heavenly Quran and they never question this, so the order of the surah would naturally also never be questioned.

Not all Muslims believed that the books we have today are perfect representations of the Heavenly Quran. The Mu'tazilites reject the idea that the Quran is uncreated or co-eternal with Allah. One of their reasons is that if Quranic verses can abrogate one another then there's no way it could be the eternal word of Allah. However, this is a VERY small and unpopular school of thought that conflicts with those who follow the traditions of the Hadith, which includes pretty much all Sunnis and Shias. Nowadays, the term "Mu'tazilite" is derogatory. That being said, The Message of The Qur'an by Muhammad Asad from the 1920's is one of the most-read versions of the Quran, regardless of alleged Mu'tazilite leanings.

This "Heavenly Quran" is also not necessarily the Quran itself, but it has historically been interpreted as such. The "concealed book" mentioned in the Quranic verses would have been revealed during the lifetime of the Prophet, before the Quran itself was put into book form. The Quran was only made into a book because those who had memorized it were dying in combat. After the Prophet passed away, Zayd bin Thabit was ordered to collect the oral and written sources and assemble it into a single volume: the Quran. So there were passages written down but they were not collected until after the life of the Prophet.

The perception of the Quran's eternality was cemented very early on by Islamic thinkers. Here's a quote from the scholar al-Tabari on the subject:

God's uncreated word however it is written or recited, whether it be in heaven or on earth, whether written on the 'guarded tablet' or on the tablets of schoolboys, whether inscribed on stone or on paper, whether memorized in the heart or spoken on the tongue; whoever says otherwise is an infidel whose blood may be shed and from whom God has dissociated Himself.

I'm sure others can provide further information on this subject. runfly24 has given a great answer - I hope my comment helps too.

It’s pretty sad that when looking for English sources on this issue, I find mostly Evangelical polemics. English Islamic scholarship is still in its infancy, but I did find one peer-reviewed paper on this topic. I have some Arabic sources I can dig up later if they would help.

But I will summarize some major points that are outlined in that paper.

The ij’ma (scholarly consensus) by Islamic scholars is that the order of the surahs is divinely inspired and authoritative. This order was put together by the Prophet himself and arranged during his lifetime. You will find authentic Hadith references in that paper that show that the Qur’an was, in fact, not only written during Muhammad’s lifetime, but was double-checked by Muhammad himself. When revelation was said to have come down to him, he would send someone to get a scribe to write the revelation down, have the scribe read what was written back to him, and correct any mistake.

As an aside and to be completely transparent, there are some differing opinions. The minority view references a Hadith that ibn Masud, one of the companions of the Prophet, had a different ordering of the text. This was a personal copy and was never claimed or thought to be authoritative. The simplest answer is that he made a mistake or didn’t care to re-write it in order. The paper speaks about this minority view and a couple others lightly.

Accounts differ on the number of scribes that wrote during the Prophet’s life ranging from 48-65. Dozens are named in the paper above. Chief among them was a young Zayd ibn Thabit who was responsible for not only writing the Quran down and cross-referencing it with the Prophet, but commissioned by Uthman ibn Affan after the Prophet’s death to compile it into one mushaf or physical binding (i.e a single book).

There is typically some confusion as to why Uthman did this or the nature of this compilation. The Caliphate did not have “a version” of the text that he preferred, ordering to burn all others, as typically charged in Christian polemics. As the Islamic community grew throughout Arabia and beyond, new converts to the faith naturally brought in their own dialects and language barriers when reading, writing, and reciting the text. Further, and what sparked Uthman to compile the text in the first place, a major battle occurred in which many of the huffaz, or people who had memorized the Qur’an in toto, were killed, and Uthman feared that the text would fall to corruption (changes) just as the Jewish and Christian texts did before it. So he commissioned Zayd to compile the Quran in written form into a single book. To add to the confusion, the Prophet revealed the Qur’an in 7 different qir’aat, recitations, that reflected the 7 most used dialects in the region. These are all authoritative and still maintained today. Uthman’s text, uses the dialect of the Quraysh which is the tribe of the Prophet. There are other points that led to confusion which I won’t go into detail here, such as the fact that the earliest written Arabic didn’t have diacritical marks to help with the pronunciation of the text. These weren’t created for several hundred years after the fact. Uthman wanted a single authoritative written copy from which all other copies would be distributed to the Kingdom’s that surrounded Arabia. But he ensured through very conservative means that this copy would indeed be the same Qur’an as recited by Muhammad.

Zayd started his compilation by first collecting what was written by himself and every other scribe he could find. This took some time as it typically wasn’t written on parchment, but on leather, scrapes of palm leaves, even bones and other materials. He then, for every verse, found two huffaz to recite the verse and cross reference it with what was written. Only then did he put it into the official compilation.

So to summarize: the Quran was not only written during the Prophet’s life, but under his supervision. The ordering of the surahs is authoritative all the way back to him according to the majority view. In another Hadith, the Prophet discusses how the Angel Gabriel would come down to him every Ramadan and review the Quran that had been revealed up to that point in time. As I’m sure you know, the tradition claims that the Quran was revealed piecemeal over the 23 year period of his prophethood. It was during these visits would the ordering of the text be revealed, and the Prophet would recite it in that order during the night prayers of Ramadan.

And not to be discounted, as I don’t think it gets enough credit, but the memorization of the Qur’an is a phenomenon not found to this extent in any other tradition. Thousands of people had memorized the Qur’an letter by letter during the Prophet’s lifetime in the correct order as it was revealed. We call this concept mutawattir. I don’t even know how to translate it. Maybe “multiply attested”, but that wouldn’t do it justice as the standard in Islamic scholarship to claim something is muttawattir is very high. But it’s the idea that the order of the Qur’an is corroborated by so many independent sources from different languages and backgrounds that they couldn’t all have conspired to lie or err. That is the biggest proof that the order goes back to Muhammad. There are people on this planet today, who can trace their memorization back to him: a student learned it from his teacher, who learned it from his teacher... who learned it from a companion, who learned it from the Prophet.

I have a couple scholarly works in mind that I believe that have been translated into English that might interest you. The compilation and authentication of the Qur’an and Hadith is an entire branch of science within Islamic scholarship, so there’s definitely more to be learned if you’re interested. I must warn: it is vast. There are scholars exclusively dedicated to isnad or the chain of transmission (person X said they heard person Y said that they heard Person Z say they heard the Prophet say...) that verify whether or not a saying can accurately be attributed to the Prophet. I believe Dr. Jonathan Brown from Georgetown would be a good starting place on this. I hope my high-level summary helps.