When did the Early Church begin and end their twice weekly fasts?

The early Christian church was effectively a sect of Judaism, so lacking any specific instruction from Jesus or the Apostles, their rules for fasting would have been the same as used by the rest of the Jewish community.

According to The Jewish Encyclopedia:

All Jewish fasts begin at sunrise and end with the appearance of the first stars of the evening, except those of the Day of Atonement and the Ninth of Ab, which last "from even till even."

There are also restrictions on the days on which one may fast:

The only fixed fast-day that may be celebrated on a Sabbath is the Day of Atonement; all the others, if they fall on a Sabbath, are postponed until the following day. Private or public occasional fasts can not be held on any of the holidays, or on a new moon, or on any of the minor festivals (see Festivals), or during the month of Nisan, or on the week-days of the festivals.

When did the Early Church begin and end their twice weekly fasts?

We know from the Didache that the Early Church fasted two days a week. But it does not expand on that nature of this particular fast as a timeframe is concerned.

The Didache also known as The Lord's Teaching Through the Twelve Apostles to the Nations is a brief anonymous early Christian treatise written in Koine Greek, dated by most modern scholars to the first century. The first line of this treatise is "The teaching of the Lord to the Gentiles (or Nations) by the twelve apostles". The text, parts of which constitute the oldest extant written catechism, has three main sections dealing with Christian ethics, rituals such as baptism and Eucharist, and Church organization. The opening chapters describe the virtuous Way of Life and the wicked Way of Death. The Lord's Prayer is included in full. Baptism is by immersion, or by affusion if immersion is not practical. Fasting is ordered for Wednesdays and Fridays.

Fasting

Chapter 8 suggests that fasts are not to be on the second day and on the fifth day "with the hypocrites", but on the fourth day and on the preparation day. Fasting Wednesday and Friday plus worshiping on the Lord's day constituted the Christian week. Nor must Christians pray with their Judaic brethren, instead they shall say the Lord's Prayer three times a day. The text of the prayer is not identical to the version in the Gospel of Matthew, and it is given with the doxology "for Yours is the power and the glory forever." This doxology derives from 1 Chronicles 29:11–13; Bruce M. Metzger held that the early church added it to the Lord's Prayer, creating the current Matthew reading. - Didache (Wikipedia)

The Benedictine Abbot Dom Prosper Guéranger, OSB in his monumental work “The Liturgical Year” (1887) has further insights to this question.

The Disciples of St. John the Baptist came, one day, to Jesus, and said to him: Why do we and the Pharisees fast often, but thy Disciples do not fast? And Jesus said to them: Can the children of the bridegroom mourn, as long as the bridegroom is with them? But the days will come, when the bridegroom shall be taken away from them, and then they shall fast. [St Matth. ix. 14,15].

In the early ages of Christianity, Fasting included also the abstaining from Wine, as we learn from St. Cyril of Jerusalem [Catech. iv], St. Basil [Homil. i. De Jejunio], St. John Chrysostom [Homil. iv. Ad populum Antioch.], Theophilus of Alexandria [Litt. Pasch, iii], and others. In the West, this custom soon fell into disuse. The Eastern Christians kept it up much longer, but even with them it has ceased to be considered as obligatory.

Lastly, Fasting includes the depriving ourselves of some portion of our ordinary food, inasmuch as it only allows the taking of one meal during the day. Though the modifications introduced from age to age in the discipline of Lent, are very numerous, yet the points we have here mentioned belong to the very essence of Fasting, as is evident from the universal practice of the Church.

It was the custom with the Jews, in the Old Law, not to take the one meal, allowed on fasting days, till sun-set. The Christian Church adopted the same custom. It was scrupulously practised, for many centuries, even in our Western countries. But, about the 9th century, some relaxation began to be introduced in the Latin Church. Thus, we have a Capitularium of Theodulph, Bishop of Orleans, (who lived at that period,) protesting against the practice, which some had, of taking their repast at the hour of None, that is to say, about three o’clock in the afternoon [Capitul. xxxix. Labb. Conc. tom. viii.]. The relaxation, however, gradually spread; for, in the 10th century, we find the celebrated Ratherius, Bishop of Verona, acknowledging, that the Faithful had permission to break their fast at the hour of None [Serm. 1, De Quadrages. D’Archery. Spicilegium, tom. ii.]. We meet with a sort of reclamation made as late as the 11th century, by a Council held at Rouen, which forbids the Faithful to take their repast before Vespers shall have begun to be sung in the Church, at the end of None [Orderic Vital. Histor., lib. iv.]; but this shows us, that the custom had already begun of anticipating the hour of Vespers, in order that the Faithful might take their meal earlier in the day.

Up to within a short period before this time, it had been the custom not to celebrate Mass, on days of Fasting, until the Office of None had been sung, (which was about three o’clock in the afternoon,) - and, also, not to sing Vespers till sun-set. When the discipline regarding Fasting began to relax, the Church still retained the order of her Offices, which had been handed down from the earliest times. The only change she made, was to anticipate the hour for Vespers; and this entailed the celebrating Mass and None much earlier in the day;- so early, indeed, that, when custom had so prevailed as to authorise the Faithful taking their repast at mid-day, all the Offices, even the Vespers, were over before that hour.

Thus we can conclude the faithful fasted twice a week on Wednesday and Friday. These two days were chosen because it was on Wednesday Our Lord was betrayed by Judas Iscariot. The Wednesday of Holy Week is still by some known as Spy Wednesday. Friday was Chosen in order to commemorate the Passion of Our Lord Jesus Christ.

We can further deduce that the faithful of the Early Church fasted from on Wednesdays meaning that they fasted from sundown Tuesday until sundown Wednesday and fasted on Fridays meaning that they abstained from food from Thursday evening (sundown) until Friday evening at sundown. The answer is that simple. They did not eat from sundown to sundown on fast days.

The Wikipedia article on the Didache states (under the heading 'Fasting'), the following :

Chapter 8 suggests that fasts are not to be on Monday and Thursday "with the hypocrites" — presumably non-Christian Jews, such as the Pharisees — but on Wednesday and Friday. Fasting Wednesday and Friday plus worshiping on Sunday constituted the Christian week. Nor must Christians pray with their Judaic brethren, instead they shall say the Lord's Prayer three times a day.