tefilla - Order of importance for the chapters of Pesukei Dizimra for someone that arrives late to Synagogue.

What is the exact order of importance of the different parts of Pizukei Dizimra when someone arrives late for Shacharit?

I am looking for something like the following:

Priority #1: Baruch Sheamar & Yistahach

Priority #2: Ahrey ...

It would help if you could also provide the reason why each part is in that priority level.

Answer

First, the entire Pesukei Dezimra can be skipped (Start with Birchas Krias Shema). Because the purpose of Pesukei Dezimra is to make the Tefilah desirable to Hashem, and praying with a Minyan accomplishes this more.

If there is time, add in Baruch Sheamar, the whole Ashrei*, and Yishtabach, because the Chachamim established the Pesukei DeZimra around Chapter 145 of Tehillim (even though the original Takana was from chapter 145 to the end of Tehillim), so doing that is a bare minimum Pesukei Dezimra.

What follows in an order of priority addition. The intention is to say the relevant passages in their proper relative order.

If there is time, add הללו אל בקדשו‏ (Tehillim 150)

If there is time, add הללו את ה' מן השמים‏ (Tehillim 148)

If there is time, say all 6 of the last chapters of Tehillim (so add in the missing three from above). The goal here being to get to the main Takanas Chachamim.

For what follows the idea is to add whole sections that are quoted over collections of verses.

If there is more time, add ויברך דוד until לשם תפארתך

Next, For those who say Hodu after Baruch Sheamar, then הודו לה' קראו until והוא רחום and then skip to the והוא רחום that is before Ashrei.

The source for the above is Shulchan Aruch HaRav O.C. 52

Kitzur Shulchan Aruch (13:7) adds in some considerations for Shabbos and Yom Tov, which are not really discussed in earlier Poskim, AFAIK, and concludes:

ואם רואה שאפלו אם יתחיל בברכת יוצר אור לא יוכל להתפלל שמונה-עשרה עם הצבור אלא אם כן ימהר, טוב לו יותר להתפלל בפני עצמו כסדר במתינות ובכונה

If he sees that even if he [skips all of the above and] starts with [the Brachos before Krias Shma of] Yotzeir he will not be able to daven Shemona Esrei unless he hurries through the davening, it is better to daven to himself [without saying Shemona Esrei with the Minayn] in order [i.e. say everything] unhurried and with proper concentration.

^{* I'm using that term loosely - the actual chapter 145 of Tehillim is what matters, the extra two verses before and one after can be skipped.}