Sleeping in

What time does Liturgy begin there in Greece? Why should we start early vs later?

Papouli,

The schedule of services throughout the year for the archdiocese of Athens are published in the dypticha each year; you can get an idea from there.

From St John Chrysostom's eighth baptismal catechesis:

16. I exhort you, therefore: let us seek the things which abide forever and never change. It was fitting, therefore, that I brought up this matter and that I exhorted all of you together, both those who have been initiated in the past and those who have just deserved the gift of baptism. On the days when we were continuously present at the tombs of the holy martyrs, we received an abundant blessing from those holy ones and enjoyed the rich benefit of their instruction. From now on, the continuity of our meetings will be broken off; hence, I must remind your loving assembly to keep ever ringing in your ears the memory of the important instruction those holy martyrs gave, and to hold spiritual things of greater importance than all the goods of this life.
17. And I urge you to show great zeal by gathering here in the church at dawn [orthron] to make your prayers and confessions to the God of all things, and to thank Him for the gifts He had already given. Beseech Him to deign to lend you from now on His powerful aid in guarding this treasure; strengthened with this aid, let each one leave the church to take up his daily tasks, one hastening to work with his hands, another hurrying to his military post, and still another to his post in the government. However, let each one approach his daily task with fear and anguish, and spend his working hours in the knowledge that at evening he should return here to the church, render an account to the Master of his whole day, and beg forgiveness for his falls. For even if we are on our guard ten thousand times a day, we cannot avoid making ourselves accountable for many and different faults. Either we say something at the wrong time, or we listen to idle talk, or we think indecent thoughts, or we fail to control our eyes, or we spend time in vain and idle things that have no connection with what we should be doing.
18. This is the reason why each evening we must beg pardon from the Master for all these faults. This is why we must flee to the loving-kindness of God and make our appeal to Him. Then we must spend the hours of the night soberly, and in this way meet the confessions of the dawn. If each of us manages his own life in this way, he will be able to cross the sea of this life without danger and to deserve the loving-kindness of the Master. And when the hour for gathering in church summons him, let him hold this gathering and all spiritual things in higher regard than anything else. In this way we shall manage the goods we have in our hands and keep them secure.

In the above we see St John's late fourth-century Constantinopolitan daily prayer programme for his flock. Notice, these are not monastics, but lay believers with all sorts of jobs. It is evident that he speaks of going to church very early in the morning, before going out to each ones' job.

Another, even earlier witness is from Egeria's early fourth-century Jerusalem pilgrimage account (24. 8-12):

But on the seventh day, the Lord's Day, there gather in the courtyard before cock-crow all the people, as many as can get in, as if it was Pascha. The courtyard is the "basilica" beside the Anastasis, that is to say, out of doors, and lamps have been hung there for them. Those who are afraid they may not arrive in time for cock-crow come early,and sit waiting there singing hymns and antiphons, and they have prayers between, since there are always presbyters and deacons there ready for the vigil, because so many people collect there, and it is not usual to open the holy places before cock-crow.
Soon the first cock crows, and at that the bishop enters, and goes into the cave in the Anastasis. The doors are all opened, and all the people come into the Anastasis, which is already ablaze with lamps. When they are inside, a psalm is said by one of the presbyters, with everyone responding, and it is followed by a prayer; then a psalm is said by one of the deacons, and another prayer; then a third psalm is said by one of the clergy, and a third prayer, and the Commemoration of All. After these three psalms and prayers they take censers into the cave of the Anastasis, so that the whole Anastasis basilica is filled with the smell. Then the bishop, standing inside the screen, takes the Gospel book and goes to the door, where he himself reads the account of the Lord's resurrection. At the beginning of the reading the whole assembly groans and laments at all that the Lord underwent for us, and the way they weep would move even the hardest heart to tears. When the Gospel is finished, the bishop comes out, and is taken with singing to the Cross, and they all go with him. They have one psalm there an a prayer, then he blesses the people, and that is the dismissal. As the bishop goes out, everyone comes to kiss his hand.
Then straight away the bishop retires to his house, and all the monazontes go back into the Anastasis to sing psalms and antiphons until daybreak. There are prayers between all these psalms and antiphons, and presbyters and deacons take their turn every day at the Anastasis to keep vigil with the people. Some lay men and women like to stay on there till daybreak, but others prefer to go home again to bed for some sleep.

From the above you can see that the first cock-crow is well before daybreak. Also, at the Anastasis we have to participation of the "spoudaioi", those monazontes referred to who were from the Sabas Lavra and were there in Jerusalem to serve the holy places, those who today are called the "phylakes tou taphou."

We can also clearly see the use of three antiphons, as is still retained in the use of the Typikon service for days without a feast, as well as the three antiphons prescribed for all great feasts. The number three, of course, is always in honor of the Holy Trinity.

Evident, also is the tradition of holding vigil on Saturday night, in honor of the resurrection, just as is still indicated in the sabaitic typika to this day. Additionally, this is why the mesonyktikon for Sunday morning, in the absence of a vigil, is in the order of the Studite/Constantinopolitan panychis (i.e., vigil).

Finally, we also see the practice of the reading of the Sunday heothinon Gospel from within the tomb by the Patriarch himself. This is why (i) we still read it today inside the bema, (ii) why, when a bishop is present, he should be present at its reading (hence, the reason why its reading is delayed till after the eighth ode of the orthros kanon) and (iii) why the reading of the orthros Gospel belongs to the clergy with the highest rank, as opposed to the Gospel of the Liturgy, which belongs to the deacons.

In any event, both examples clearly show how the early christians, even if they were not making vigil, came to church at first cock-crow. Anyone who has lived in the country knows that this is at the very first breaking of the darkness, say around 3-4AM.