Is it true that early Christians were baptised naked?

Quick answer: Yes, nude baptism was practiced in Ancient times.

From A Dictionary of Christian Antiquities p 160 ed W Smith & S Cheetam (1875)

A comparison of all the evidence leads to the conclusion that the catechumens entered the font in a state of absolute nakedness. See particularly St Cyril, Hieros. Myst. Catech. ii ad init; St Ambrose, Serm. xx (Opp. t.v. p. 153, Paris 1642)and Enarrat. in Ps lxi 32 (BB t.i.p. 966); St Chrysostom, ad Illum. Cat. i (Migne, tom. ii. p 268). Possibly a cincture of some kind (quo pudori consuleretur) may have been worn, as indicated in some medieval works of art.

St. Cyril of Jerusalem (4th Century)

Therefore, I shall necessarily lay before you the sequel of yesterday's Lecture, that ye may learn of what those things, which were done by you in the inner chamber(2), were symbolical. 2. As soon, then, as ye entered, ye put off your tunic; and this was an image of putting off the old man with his deeds(3). Having stripped yourselves, ye were naked; in this also imitating Christ, who was stripped naked on the Cross, and by His nakedness put off from Himself the principalities and powers, and openly triumphed over them on the tree(4). For since the adverse powers made their lair in your members, ye may no longer wear that old garment; I do not at all mean this visible one, but the old man, which waxeth corrupt in the lusts of deceit(5). May the soul which has once put him off, never again put him on, but say with the Spouse of Christ in the Song of Songs, I have put off my garment, how shall I put it on(6)? O wondrous thing! ye were naked in the sight of all, and were not ashamed(7); for truly ye bore the likeness of the first-formed Adam, who was naked in the garden, and was not ashamed. 3. Then, when ye were stripped, ye were anointed with exorcised oil(8), from the very hairs of your head to your feet, and were made partakers of the good olive-tree, Jesus Christ. For ye were cut off from the wild olive-tree(9), and grafted into the good one, and were made to share the fatness of the true olive-tree.

Bishop Hippolytus of Rome:

21 1 At the hour in which the c**k crows, they shall first pray over the water. 2 When they come to the water, the water shall be pure and flowing, that is, the water of a spring or a flowing body of water. 3 Then they shall take off all their clothes.The children shall be baptized first. All of the children who can answer for themselves, let them answer. If there are any children who cannot answer for themselves, let their parents answer for them, or someone else from their family. 5 After this, the men will be baptized. Finally, the women, after they have unbound their hair, and removed their jewelry. No one shall take any foreign object with themselves down into the water. ...

9 When the elder takes hold of each of them who are to receive baptism, he shall tell each of them to renounce, saying, "I renounce you Satan, all your service, and all your works." 10 After he has said this, he shall anoint each with the Oil of Exorcism, saying, "Let every evil spirit depart from you." 11 Then, after these things, the bishop passes each of them on nude to the elder who stands at the water. They shall stand in the water naked. A deacon, likewise, will go down with them into the water. (Hippolytus. "Apostolic Traditions" of Hippolytus, 21:1-11. Translated by Edgecomb, Kevin P. Derived from Bernard Botte (La Tradition Apostolique. Sources Chretiennes, 11 bis. Paris, Editions du Cerf, 1984) and of Gregory Dix ^{(The Treatise on the Apostolic Tradition of St. Hippolytus of Rome, Bishop and Martyr. London: Alban Press, 1992)}

My understanding from seminary was that, for modesty, women were baptised in darkness. Eventually, for the same reason, the practice fell out of favor.