The Roman tradition is early but it is tradition, and its parts do not all stand equally.
That Peter died a martyr is attested indirectly within the New Testament, in the closing chapter of John, where Jesus tells him he will stretch out his hands and be carried where he does not wish to go, and the text glosses this as indicating the death by which he would glorify God. That is written from the far side of the event. Clement of Rome, writing near the end of the first century, refers to Peter's suffering and death without saying where. Ignatius of Antioch, early in the second century, writes to the Romans in terms that assume Peter and Paul had authority there. By the later second century the association with Rome is firm, and a Roman presbyter named Gaius points visitors to memorials of the two apostles on the Vatican hill and on the Ostian road.
The inverted crucifixion is later and thinner. It appears in the Acts of Peter, an apocryphal romance of the later second century, where Peter asks to be crucified head downward because he is unworthy to die as his master did. Nothing earlier reports it. It is a good story with a clear theological motive and no independent corroboration, and it should be described as what it is.
The bones are a separate question. Excavations under St Peter's from 1939 uncovered a first-century necropolis and a second-century monument, and a set of bones from a niche near it was declared by Paul VI in 1968 to be identified in a manner the church considers convincing. The archaeology of the cemetery and the shrine is solid; the identification of the individual remains is not, and the archaeologist who led the dig disputed it.