Was the Gathering Relics of Holy Men in 1st Century common in the Early Church? I am hoping to understand or know if Jesus' apostles would have had gathering relics of something like the crucifixion in mind at some point when it as still possible to locate them (ex: the true cross, crown of thorns, nails), such as a few days following the crucifixion?

Was this something Jews had a habit of doing for Holy men during Christ's time on earth? Unless this was a cultural/religious practice, I have some skepticism on the likelihood that this would have been a priority.
Neither Britannica nor Wikipedia list any form of relics within Judaism.

Collecting relics was effectively never done by the Israelites or Jews.  To them, only God (and God through his agents) has supernatural power.  Believing that relics or other objects have supernatural power, in principle, violates the first and second commandments, "Thou shalt have no other gods before me" and "Thou shalt not make unto thee any graven image".  It in effect turns the object into a form of god.

Ancient Greeks though, did believe in the power of relics, and this belief passed on to the Roman Empire and the Holy Roman Empire.

It wasn't until the Roman Church became the official Christian form of Christianity that anyone thought to collect Christian relics.

Constantine's mother, Empress Helena, was particularly interested in relics, artefacts, and locations.  She herself made a pilgrimage to Palestine and other provinces where she built churches on holy sites, including the location of Moses's burning bush.  At the site of Jesus's tomb, she had three crosses excavated and used a miracle to determine which one was the True Cross.  She also discovered the crucifixion nails and placed one in Constantine's helmet and one in his horse's bridle.  She is also thought to have found Jesus's Holy Tunic.

Helena explicitly sent her men to locate holy relics. Given what failure would have meant, all these missions were of course successful.  Other finds included the rope used to tie Jesus to the cross, more pieces of the cross, and more of the Holy Tunic.

Centuries later, many Crusaders returned with "genuine" relics they had liberated from the holy land.  Some were naïve dupes, while others were scoundrels in their own right.  The number of nails and pieces of the true cross that were brought back would have been sufficient to build a church or two.

In reality, the chance that anyone would have preserved Jesus's cross or its nails, his crown of thorns, his blood, etc. are extremely close to zero.