While Paul had to remain in Rome, Peter would not have had to and no doubt would have left there at that time, seeing the trouble that was coming.
He would have left Italy, gone back to Asia Minor, then on to Parthian Babylon where he would then write his first epistle in 65 AD, and probably his second in 66AD, probably dying there shortly thereafter.
The apostles were missionaries, not settlers or bishops, and Peter and the other apostles lived by the words of Jesus recorded in Matthew 10:23: "But when they persecute you in this city, flee into another", until, of course, their end had come.
After Paul's first imprisonment I believe he traveled to Spain...and then on to England before returning to Asia Minor. There is a gap between the two imprisonments and scripture does speak of him wanting to go to Spain [Romans 15:24 and Romans 15:28]. We know that his first visit to Rome was in custody so it is reasonable to believe he went on to Spain after being released. The "Sonnini Manuscript" tells of Paul's visit to Britain and Spain. This would have been the only time to accomplish this journey and I believe would have left him little time to meet Simon Peter in Rome.