Handwaving away the OP (as theologically the Papacy exists at least in the second century, see Irenaeus' letter to pope victor on the date of easter), let's speculate that the church in rome is totally destroyed, Peter himself dies in Jerusalem and this gets mentioned in a few of Paul's letters so there is not dispute as to where Peter died in the Christian world. Then, Jerusalem gets annhilated as per OTL and only ANtioch could claim to be a historical patriarchate (as Alexandria only became big as a legend of Peter sending Mark to Egypt made it a tangentially petrine patriarchate.)
Here's what I think happens.
First, as per 1 COr 11:3, Matthew 16:18, and other passages, the idea of a hierarchical Christian organization is theologically necessary. So, we are going to get a monarchic epsicopacy anyway.
Second, I personally believe (as an Orthodox believer) that Patriarchates are political evolutions. Securlarly speaking, it would be almost impossible to deny this. Canon 28 of Chalcedon offers this rationale, as well as the fact that Rome and Alexandria retroactively became Petrine as clearly they had Bishops BEFORE Peter was ever there (or his representative to Alexandria), as well as Antioch being Petrine. it seems to me that being a "Petrine" patriarchate is equivalent to having the word "Guards" thrown in front of your army in the USSR. It was in reality an honorific without a literal historical reality.
Third, accepting the previous 2 premises, we are still going to end up with similar patriarchates. Rome, as the seat of the empire, will be important. Constantinople will also be elevated. Perhaps Ephesus (Johnnine primacy) and Antioch may have a little more historical claims, but they will still be politically sidelined for Rome and COnstantinople (and lesser extent Alexandria, see Edict of Thessalonica 383). So, I honestly think we have schisms along the same lines we have today.
What does change is the rationale for Roman primacy. Ultimately, I think Roman Catholics would be making Orthodox arguments: "before the schism, we were recognized as the top patriarchate and until there is another council we still are." What Rome would lack is a metaphysical chrism that truly sets them apart. but, as we see with Constantinople and Ukraine today, a church with a few thousand of its own believers due to canonical precedent can make claims to jurisdictional supremacy. so, I think we have what we see today, with the argument phrased in much more canonical terms instead of "peter did this and that."