The foundational principle underlying all the rulings is that Pesukei DeZimra were instituted only as preparation for prayer, so it is better to skip portions of them and pray Shemonah Esrei with the congregation — as the Shulchan Arukh HaRav (Orach Chayim 52:1–7) reasons explicitly: "Pesukei DeZimra were instituted only for the sake of the prayer, that it should be accepted... and if so, it is better to pray with the congregation, for then his prayer is certainly accepted."
The practical hierarchy of what to say when time is short is spelled out in the Shulchan Arukh (Orach Chayim 52:1–5): if the congregation is near the end of Pesukei DeZimra, one says Baruch She'amar, then Tehillah LeDavid, then the two Hallelu passages, then Yishtabach, then Yotzer and Keriat Shema with its blessings, and then prays with the congregation; if there is less time, one skips Hallelu Et Hashem Min Hashamayim; and if even less time remains, one says only Baruch She'amar, Tehillah LeDavid, and Yishtabach.
The Chayyei Adam (19:1–11) clarifies that the essential core of Pesukei DeZimra is Tehillah LeDavid through "va'anachnu," and that one must never recite Baruch She'amar or Yishtabach without at least one mizmor between them — those who begin directly from Yishtabach recite a blessing in vain.
If the congregation has already begun Yotzer and there is no time for even an abridged Pesukei DeZimra, the Arukh HaShulchan (Orach Chayim 52:1–10) rules that one reads Keriat Shema and its blessings with the congregation and prays with them, and then recites all of Pesukei DeZimra afterward without the opening and closing blessings.