Part II here : https://petercanisiusmichaeldavidkang.com/2013/03/09/my-guide-to-the-papal-conclave-part-ii-from-the-beginning-to-the-election-of-the-new-pope/
1. After the new Pope had accepted his election as Pope and leader of the Roman Catholic Church, there are several scenarios :
a. If he is already a bishop : The new Pope will immediately be the Bishop of Rome and leader of the Universal Church, the Successor of St. Peter the Apostle. Then he will immediately accept the homage of the Cardinal-electors gathered in the Conclave, after he changes his cardinal’s robes for the new, white papal robes in the Room of Tears.
b. If he is not yet a bishop : The new Pope will need to be immediately ordained a bishop following the proper order of the Ordination of bishops, and the senior Cardinal Bishop by seniority in the Conclave should be the principal consecrator. The new Pope can only receive homage of the Cardinals after his ordination to the episcopate (after he had been ordained a bishop).
2. The new Pope then leaves the Sistine Chapel main hall where the voting took place, into the special ‘Room of Tears’, where three sets of differently-sized white papal robes had been prepared for him to adjust to his size. The new Pope changes from his cardinal’s red robes into the white papal robes and then return to the main hall of the Sistine Chapel.
3. The new Pope, after dressed in the white papal robes (with the red mozzetta and the large state stole) will then receive the homage of all the Cardinals gathered in the Sistine Chapel.
4. New addition in 2013 Conclave : After the homage, the new Pope will proceed with all the Cardinal-electors from the Sistine Chapel to the Pauline Chapel. The new Pope will enter the Pauline Chapel alone, and pray before the Blessed Sacrament in the Chapel.
5. In the meanwhile, the most senior Cardinal in the order of deacons, that is the Cardinal Protodeacon (Cardinal Jean-Louis Tauran for the 2013 conclave) will then appear at the Loggia or the balcony of St. Peter’s Basilica to deliver the announcement of the election of a new Pope.
The formula :
The Cardinal Protodeacon will first greet the assembled people in St. Peter’s Square and throughout the world (done in 2005 conclave in different languages) :
1. Italian : Fratelli e sorelle carissimi
2. Spanish : Queridísimos hermanos y hermanas
3. French : Bien chers frères et sœurs
4. German : Liebe Brüder und Schwestern
5. English : Dear brothers and sisters
Then he continues :
Annuntio vobis gaudium magnum
I announce to you a great joy
We have a Pope!
Eminentissimum ac Reverendissimum Dominum
The most eminent and reverend Lord
Dominum (New Pope’s first name/baptismal name) Sanctae Romanae Ecclesiae Cardinalem (New Pope’s surname)
Lord (New Pope’s first name/baptismal name) Cardinal (New Pope’s surname) of the Holy Roman Church
Qui sibi nomen imposuit (New Pope’s chosen regnal name and number)
Who takes for himself the name of (New Pope’s chosen regnal name and number)
5. After this, after a moment, the new Pope, led by a processional crucifix, made his appearance at the same balcony of St. Peter’s Basilica. He then may deliver his first address as Pope to the people gathered at the Square, and to the world.
6. After the address, then the new Pope will give his Urbi et Orbi special blessing (For the city of Rome (Urbi) and the world (Orbi)), with which is attached plenary indulgence, that is the forgiveness of all temporal sins, providing the people participates in full faith and repentance.
The text of the Urbi et Orbi blessing (Latin):
Sancti Apostoli Petrus et Paulus: de quorum potestate et auctoritate confidimus ipsi intercedant pro nobis ad Dominum.
Precibus et meritis beatæ Mariae semper Virginis, beati Michaelis Archangeli, beati Ioannis Baptistæ, et sanctorum Apostolorum Petri et Pauli et omnium Sanctorum misereatur vestri omnipotens Deus; et dimissis omnibus peccatis vestris, perducat vos Iesus Christus ad vitam æternam.
Indulgentiam, absolutionem et remissionem omnium peccatorum vestrorum, spatium verae et fructuosae poenitentiæ, cor semper penitens, et emendationem vitae, gratiam et consolationem Sancti Spiritus; et finalem perseverantiam in bonis operibus tribuat vobis omnipotens et misericors Dominus.
Et benedictio Dei omnipotentis, Patris et Filii et Spiritus Sancti descendat super vos et maneat semper.
Text of Urbi et Orbi blessing (English)
May the Holy Apostles Peter and Paul, in whose power and authority we have confidence, intercede on our behalf to the Lord.
Through the prayers and merits of the Blessed Mary ever Virgin, of Blessed Michael the Archangel, of Blessed John the Baptist, and of the Holy Apostles Peter and Paul, and of all the saints, may Almighty God have mercy on you, and with your sins forgiven, may Jesus Christ lead you into everlasting life.
May the Almighty and merciful Lord grant you indulgence, absolution, and remission of all your sins, time for a true and fruitful penance, an always repentant heart and amendment of life, the grace and consolation of the Holy Spirit, and final perseverance in good works.
And may the blessing of Almighty God, the Father, the Son, and the Holy Spirit, descend on you and remain with you always.
7. Then the new Pope takes his leave of the people, and return inside St. Peter’s Basilica. The long process of electing a new Pope is over, and the new Pope is now in charge in leading God’s Church in his given mission as the Successor of St. Peter the Apostle.
8. The Papal Inauguration Mass will take place several days after the election (does not have to be on a Sunday), to allow for the foreign dignitaries invited to the event to be able to arrive, to attend the Papal Inauguration Mass. The new Pope will receive his signs of office in this Inauguration, namely, the pallium and the Ring of the Fisherman. The Papal Inauguration most likely will take place at St. Peter’s Square.
9. The new Pope will take possession of his Cathedral, in an enthronement ceremony at the Cathedral of the Diocese of Rome (The Pope is the Bishop of Rome, and contrary to popular perceptions, St. Peter’s Basilica is not a Cathedral). The Cathedral of Rome is the Archbasilica of St. John Lateran
Complete name (Latin) : Archibasilica Sanctissimi Salvatoris et Sanctorum Ioannes Baptista et Evangelista in Laterano, Omnium urbis et orbis Ecclesiarum Mater et Caput
Complete name (English) : Archbasilica of the Most Holy Saviour, Saint John the Baptist, and Saint John the Evangelist in Lateran, Head and Mother Church of All the City and the Whole World