The old man (the old or first nature), the body of sin and the flesh are all the same thing but viewed in different aspects.
The old man or old nature is that which is of Adam corporately as Paul discusses in Romans 5:12 -21, a matter of headship and progeny.
The body of sin views the natural part of humanity, the flesh and blood and bones, as a container within which dwells sin, one's own supply of the sin of the world. Just as we speak of a 'body' of water, a contained area of a discrete amount of the fluid substance, so the 'body' of sin is one's own contained area of that which is common to all.
Then, the flesh itself is that physical thing with which sin is associated physically. Sin dwells in flesh - it does not exist anywhere else (apart from its original source, that is).
The answer to all of this is that Jesus Christ was made sin, II Corinthians 5:21. He became the containment of sin, in his hours upon the cross. 'Made' is not quite correct as the Greek poieo is a broad concept covering both 'make' and 'do'. 'Effected' or 'caused' is closer to the concept of poieo.
Once he died, sin was gone, lost in his death. The old man is crucified with Christ, the body is dead because of sin. Sin is destroyed.