Short answer
It depends what you mean. As the community following Prophet Muhammad ﷺ, Islam began in the 7th century, about six centuries after Jesus. But Muslims believe Islam, meaning submission to the one God, is the original religion taught by every prophet from Adam through Abraham, Moses and Jesus, so in that sense it is the oldest faith.
Muslims believe the religion (din) of all the prophets was one, Islam, while their laws (shari'a) differed. The shariah of Muhammad ﷺ is the final law and replaces the earlier ones.
The four schools agree on this.
Explanation
Historically, the first revelation came to Prophet Muhammad ﷺ in about 610 CE in Makkah. Christianity began in the first century, so as organised communities, Christianity came first. See did Muhammad come before or after Jesus.
The Quran sees it differently. It says the religion in the sight of Allah is Islam, and that Allah ordained for Muslims the same religion He enjoined on Noah, Abraham, Moses and Jesus. It says Abraham was neither a Jew nor a Christian but a Muslim, submitting to Allah, and that the disciples of Jesus declared: we are Muslims. In Arabic, muslim simply means one who submits to God.
The Prophet ﷺ explained it with a family picture: the prophets are brothers from one father with different mothers, and their religion is one. The core never changed: worship the one God, believe in the Last Day, do good. The laws did change from prophet to prophet, which the Quran confirms: to each of you We prescribed a law and a way.
This is why Muslims do not call their faith Muhammadanism. They do not worship Muhammad ﷺ; they follow the final messenger of a very old message. See was Jesus a Muslim, do Muslims pray to Muhammad and what is Islam.
The evidence
Quran, Surah Aal-i-Imraan 3:67
مَا كَانَ إِبْرَٰهِيمُ يَهُودِيًّۭا وَلَا نَصْرَانِيًّۭا وَلَٰكِن كَانَ حَنِيفًۭا مُّسْلِمًۭا وَمَا كَانَ مِنَ ٱلْمُشْرِكِينَ ﴿٦٧﴾
Abraham was neither a Jew nor a Christian, but he was one inclining toward truth, a Muslim [submitting to Allah]. And he was not of the polytheists.
Abraham followed the religion of submission to God.
Quran, Surah Ash-Shura 42:13
۞ شَرَعَ لَكُم مِّنَ ٱلدِّينِ مَا وَصَّىٰ بِهِۦ نُوحًۭا وَٱلَّذِىٓ أَوْحَيْنَآ إِلَيْكَ وَمَا وَصَّيْنَا بِهِۦٓ إِبْرَٰهِيمَ وَمُوسَىٰ وَعِيسَىٰٓ ۖ أَنْ أَقِيمُوا۟ ٱلدِّينَ وَلَا تَتَفَرَّقُوا۟ فِيهِ ۚ كَبُرَ عَلَى ٱلْمُشْرِكِينَ مَا تَدْعُوهُمْ إِلَيْهِ ۚ ٱللَّهُ يَجْتَبِىٓ إِلَيْهِ مَن يَشَآءُ وَيَهْدِىٓ إِلَيْهِ مَن يُنِيبُ ﴿١٣﴾
He has ordained for you of religion what He enjoined upon Noah and that which We have revealed to you, [O Muhammad], and what We enjoined upon Abraham and Moses and Jesus - to establish the religion and not be divided therein. Difficult for those who associate others with Allah is that to which you invite them. Allah chooses for Himself whom He wills and guides to Himself whoever turns back [to Him].
All prophets were given the same religion.
Quran, Surah Al-Maaida 5:48
وَأَنزَلْنَآ إِلَيْكَ ٱلْكِتَٰبَ بِٱلْحَقِّ مُصَدِّقًۭا لِّمَا بَيْنَ يَدَيْهِ مِنَ ٱلْكِتَٰبِ وَمُهَيْمِنًا عَلَيْهِ ۖ فَٱحْكُم بَيْنَهُم بِمَآ أَنزَلَ ٱللَّهُ ۖ وَلَا تَتَّبِعْ أَهْوَآءَهُمْ عَمَّا جَآءَكَ مِنَ ٱلْحَقِّ ۚ لِكُلٍّۢ جَعَلْنَا مِنكُمْ شِرْعَةًۭ وَمِنْهَاجًۭا ۚ وَلَوْ شَآءَ ٱللَّهُ لَجَعَلَكُمْ أُمَّةًۭ وَٰحِدَةًۭ وَلَٰكِن لِّيَبْلُوَكُمْ فِى مَآ ءَاتَىٰكُمْ ۖ فَٱسْتَبِقُوا۟ ٱلْخَيْرَٰتِ ۚ إِلَى ٱللَّهِ مَرْجِعُكُمْ جَمِيعًۭا فَيُنَبِّئُكُم بِمَا كُنتُمْ فِيهِ تَخْتَلِفُونَ ﴿٤٨﴾
And We have revealed to you, [O Muhammad], the Book in truth, confirming that which preceded it of the Scripture and as a criterion over it. So judge between them by what Allah has revealed and do not follow their inclinations away from what has come to you of the truth. To each of you We prescribed a law and a method. Had Allah willed, He would have made you one nation [united in religion], but [He intended] to test you in what He has given you; so race to [all that is] good. To Allah is your return all together, and He will [then] inform you concerning that over which you used to differ.
The laws differed while the religion stayed one.
Sahih al-Bukhari 3443
Sahih... الأنبياء إخوة لعلات أمهاتهم شتى ودينهم واحد ...
"The prophets are brothers from one father; their mothers are different and their religion is one."
All prophets taught one religion.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.