The following narrations found in history from both Shia and Sunni sources seem to indicate that Rabi al-Awwal was the first month of the Islamic calendar as established by Prophet Muhammad (saww) and this was later changed to Muharram by Umar (la).

Shia Sources:

وقال ابن شهراشوب: ”قال الطبري ومجاهد في تاريخهما: جمع عمر بن الخطاب الناس يسألهم من أي يوم نكتب؟ فقال علي عليه السلام: من يوم هاجر رسول الله صلى الله عليه وآله وترك أهل الشرك. فكأنه أشار إلى أن لا تبتدعوا بدعة، وتؤرّخوا كما كانوا يكتبون في زمان رسول الله صلى الله عليه وآله، لأنه لمّا قَدِمَ النبي صلى الله عليه وآله المدينة في شهر ربيع الأول أمر بالتاريخ، فكانوا يؤرخون بالشهر والشهرين من مقدمه إلى أن تمّت له سنة.

Ibn Shahr Ashoob says, "al-Tabari and Mujahid say in their books of History":

"Umar bin Khattab gathered the people, he asked them from which date should we write the calendar (start of the calendar)? Imam Ali (as) said, "From the day the Prophet (saww) did Hijrah (migrated from Mecca to Medina) and left the people of shirk (Mushrikeen). With this he suggested that do not initiate a Bidah (innovation) and leave it as it was written down in the time of the Prophet (saww), since it was when the Prophet (saww) set forth to Medina in the month of Rabi al-Awwal, he commanded the start of the date (start of the calendar). But they put it back by a month and two months in advance to what was the (start of) year."

[Source: Manaqib - ibn Shahr Ashoob, Vol. 1, Pg. 338]

Mukhalifeen (Sunni) Source:

حَدَّثَنَا عَبْدُ اللَّهِ بْنُ مَسْلَمَةَ، حَدَّثَنَا عَبْدُ الْعَزِيزِ، عَنْ أَبِيهِ، عَنْ سَهْلِ بْنِ سَعْدٍ، قَالَ مَا عَدُّوا مِنْ مَبْعَثِ النَّبِيِّ صلى الله عليه وسلم وَلاَ مِنْ وَفَاتِهِ، مَا عَدُّوا إِلاَّ مِنْ مَقْدَمِهِ الْمَدِينَةَ

Narrated Sahl bin Sa'd:

The Prophet's companions did not take as a starting date for the Muslim calendar, the day, the Prophet (saww) had been sent as an Apostle or the day of his death, but the day of his arrival at Medina (i.e. the month in which the Prophet migrated from Mecca to Madina).

[Source: Saheeh al-Bukhari, Book 63, Hadith 160 - Online Link]

وقال القسطلاني: ”وأمر صلى الله عليه وسلم بالتاريخ فكُتِبَ من حين الهجرة، وقيل: إن عمر أول من أرّخ وجعله من المحرّم

And al-Qastalani said, "The Prophet (saww) ordered for the date (calendar) to start and its was written from when He did the Hijrah (migration from Mecca to Medina). And its is said that verily Umar was the first who set the date and made it to Muharram."

[Source: Al-Mawahib by al-Qastalani, Vol. 1, Pg. 67]

وقال الصاحب بن عباد: ”ودخل المدينة يوم الإثنين لاثني عشرة خلت من ربيع الأول، وكان التاريخ من ذلك، ثم رُدَّ إلى المحرّم

And al-Sahib bin Abad said, "And He entered Madina on a Monday after 12 days had passed in Rabi al-Awwal, and the start of the date (start of the calendar) was from there, then it was changed to Muharram."

[Source: Unwan al-Maaraif by Sahib bin Abad, Pg. 11]