In answer to the OP, no, the delay in burial does not attest to the fact of no close relatives. Rather, it attests to two other things. One was the fact that Jesus had been tried as a criminal, found guilty, and hung on the cross by some very powerful people. Who would stand up? Two was the fact that the authorities did not know Jesus had actually died at 3pm.
Who stood up?
Obviously, Jesus had a family.
Is not this [Jesus] the carpenter, the son of Mary, the brother of James, and Joses, and of Juda, and Simon? and are not his sisters here with us? And they were offended at him. Mk 6:3
He also had male and female disciples or followers some of whom were relatives.
And many women were there beholding afar off, which followed Jesus from Galilee, ministering unto him: Among which was Mary Magdalene, and Mary the mother of James and Joses, and the mother of Zebedee's children.
Now the names of the twelve apostles are these; The first, Simon, who is called Peter, and Andrew his brother; James the son of Zebedee, and John his brother; Philip, and Bartholomew; Thomas, and Matthew the publican; James the son of Alphaeus, and Lebbaeus, whose surname was Thaddaeus; Simon the Canaanite, and Judas Iscariot, who also betrayed him. Mt 10:2-4
But they [Mary and Joseph], supposing him [Jesus] to have been in the company, went a day's journey; and they sought him among their kinsfolk and acquaintance. Lk 2:44
And of course there is Nicodemus and Joseph of Arimathaea. They are not related by blood, but by hope. They stood up against the authorities of Rome and of the Sanhedrin.
And after this Joseph of Arimathaea, being a disciple of Jesus, but secretly for fear of the Jews, besought Pilate that he might take away the body of Jesus: and Pilate gave him leave. He came therefore, and took the body of Jesus. And there came also Nicodemus, which at the first came to Jesus by night, and brought a mixture of myrrh and aloes, about an hundred pound weight. Jn 19:38-39
Death on the cross
That Jesus died about 3pm is not in question. The ninth hour in Jewish time is 3pm, counting 9 hours from 6am (sunrise).
And when the sixth hour was come, there was darkness over the whole land until the ninth hour. ... And Jesus cried with a loud voice, and gave up the ghost. Mk 15:33, 37
What we find out, however, is the authorities' surprise that Jesus had died so quickly. They would not fail at their duty.
And now when the even was come, because it was the preparation, that is, the day before the sabbath, Joseph of Arimathaea, an honourable counsellor, which also waited for the kingdom of God, came, and went in boldly unto Pilate, and craved the body of Jesus. And Pilate marvelled if he were already dead: and calling unto him the centurion, he asked him whether he had been any while dead. And when he knew it of the centurion, he gave the body to Joseph. Mk 15:42-45
The phrase "when the even was come" tells us that it was past sunset. The new day had begun. The Romans did not care about Sabbaths and Feasts. The Jews did.
The Jews therefore, because it was the preparation, that the bodies should not remain upon the cross on the sabbath day, (for that sabbath day was an high day,) besought Pilate that their legs might be broken, and that they might be taken away. Jn 19:31
Should not remain on the cross on the sabbath day.
His body shall not remain all night upon the tree, but thou shalt in any wise bury him that day; (for he that is hanged is accursed of God;) that thy land be not defiled, which the LORD thy God giveth thee for an inheritance. Deut 21:23
Jesus died about 3pm, yet remained on the cross. In the evening, Joseph must have been at the cross. He sees that the prisoners are dead by the breaking of legs, but of Jesus by the piercing that He was already dead. Joseph goes to Pilate who confirms these things. Pilate grants the body to Joseph who carries out the burial with Nicodemus.
The so-called delay was due simply to the fact Jesus gave up the ghost "early". Once Joseph found out Jesus was dead, he immediately did what was necessary to effect the burial.
Lastly, it is important to note that Joseph was a member of the Sanhedrin. He went against their authority.
And, behold, there was a man named Joseph, a counsellor; and he was a good man, and a just: (The same had not consented to the counsel and deed of them;) he was of Arimathaea, a city of the Jews: who also himself waited for the kingdom of God. This man went unto Pilate, and begged the body of Jesus. Lk 23:50-52
There was no delay when the facts were known.