Waiting for Minyan by Yishtabach-Minyan arrived after Yishtabach:
Where is the Chazan to wait: In the event that a Minyan is not yet available in Shul, the Chazan is to wait [between Az Yashir and] Yishtabach [until the arrival of the tenth man]. He is to wait silently, without saying anything, until the tenth man arrives, upon which he is to say Yishtabach and Kaddish. The Chazan may wait [between Az Yashir and Yishtabach] even for a period of thirty minutes, which is the amount of time it takes to complete the entire Pesukei Dezimra, from Baruch Sheamar until the end of Yishtabach. [Practically, the Chazan is to wait by the end of Az Yashir, by Vehaya Hashem Limelech, and then recite from there aloud when the Minyan arrives.]
The law if the Chazan already said Yishtabach: If the Chazan already recited Yishtabacha prior to the arrival of the Minyan then after the Minyan arrives he is to repeat part of Pesukei Dezimra and then say [the half] Kaddish [and Barchu]. [Initially three verses are to be repeated aloud, although Bedieved it suffices to say two verses aloud. It is not necessary for the entire Minyan to repeat three verses, but simply the Chazan alone.] If the Chazan [and others] do not wish to repeat the verses, then Kaddish is not to be recited and they are rather to say Barchu without Kaddish.
Where is the rest of the Minyan to wait: In the event that a Minyan is not yet available, and one must wait until the arrival of the tenth man, the individuals are not to wait between Az Yashir and Yishtabach but are rather to recite Yishtabach and then wait for the Minyan to arrive. [Alternatively, they are to wait in the middle of Az Yashir and then complete Az Yashir and say Yishtabach when the Minyan arrives.]
 Admur 53:3; Rama 53:3; M”A 53:5; Maharil Tefila 7; Chesed Lealafim 54:3; Kitzur SHU”A 15:1; Kaf Hachaim 55:5; Ketzos Hashulchan 18:10; Piskeiy Teshuvos 53:4
 Admur ibid; M”A ibid; Maharam Mintz 8
Other opinions: Some Poskim rule that the Chazan may wait up until a half hour before Yishtabach, however more than a half hour he may not wait and is rather to begin Yishtabach right away, and wait between Yishtavbach and Yotzer. [Kitzur SHU”A ibid]
 Admur ibid
The reason: Even if the Chazan waits for more than a half hour between Az Yashir and Yishtabach, nevertheless, he is not required to repeat Pesukei Dezimra, as a delay within Davening only requires a repetition if one made a delay due to Oness [i.e. in bathroom] and not if the delay was caused willingly, as explained in chapter 65:1. [Admur ibid]
 Yifei Laleiv 53:5; Mishmeres Shalom Minhagim 8; Piskeiy Teshuvos ibid
The reason: In order to approximate the end of Pesukei Dezimra to Yishtabach, as brought in Poskim in coming footnotes.
 So rule regarding if made a delay between Yishtabach and Kaddish: Admur 54:4; Rama 54:3; Sefer Hamachkim of Rav Nasan Bar Yehuda; Orchos Chaim; Kol Bo 6; So rule regarding our case if no Minyan was available by Yishtabach and then a Minyan arrived: Derech Hachaim 30; Kitzur SHU”A 15:1; M”B 53:11 [however see 55:2 regarding if Minyan immediately arrived]; Ketzos Hashulchan 18:10; Piskeiy Teshuvos ibid; Regarding the general need to repeat verses with the Minyan before reciting Kaddish see: M”A 54:3; 69:4; 234:1; Elya Raba 55:3; Chesed Lealafim 54:3; Chayeh Adam 30:7; M”B 54:8 and 55:2 [concludes that Achronim rule like M”A ibid]; Kaf Hachaim 55:5 and 41; Ketzos Hashulchan 16:2 footnote 6
Other opinions: Some Poskim rule it is not necessary to have ten people by the recital of the verses so long as there are ten people by the Kaddish, and thus there is no need to repeat anything aloud when the tenth man arrives. [Taz 55:3, brought in M”B ibid and Kaf Hachaim 55:41]
If the Minyan arrived immediately after Yishtabach: Some Poskim rule that so long as the Minyan arrived immediately after Yishtabach, then verses do not need to be repeated, and it is only to be repeated if an interval was made. [P”M 55 M”Z 3; M”B 55:2; Piskeiy Teshuvos 53:4] This is not the simple implication of the Poskim ibid
 The reason: As Kaddish may never be said unless it was preceded by verses of praise. [Poskim ibid]
 Admur ibid and Kitzur SHU”A ibid writes a few verses; Chesed Lealafim 55:3 and Kaf Hachaim 55:5 write 2-3 verses; M”B 53:11 and Ketzos Hashulchan ibid writes three verses; Piskeiy Teshuvos 55:3 footnote 17 in name of Beis David 30 writes as we stated above.
 M”B 54:3; Implication of all Poskim ibid
 M”A 53:4; M”B 53:9; Aruch Hashulchan 53:4; Ketzos Hashulchan 18:10; Darkei Chaim Veshalom 131; Piskeiy Teshuvos 53:4
 The reason: As one is to recite Yishtabach immediately after Az Yashir, without making an interval in-between. [Poskim ibid] Seemingly this is because Yishtabach is the after blessing of Pesukei Dezimra and it hence must be approximated to the Psalms of Pesukei Dezimra. Alternatively, this is based on Kabala and is done in order to unify Hashem’s name of Shakaiy and approximate the word Echad to Yishtabach. [See Shalmei Tzibur]