The gemara[1] states that one who recites Kri’at Shema without tefillin is like one who says false testimony about himself. Rabbeinu Yona[2] explains that since Kri’at Shema contains the commandment to don tefillin, it is problematic to read about that mitzva and not fulfill it. There are several sources that say that, for positive and/or to avoid negative issues, it is important to also have tefillin on during Shemoneh Esrei.[3] Therefore, it is common that those who are kind enough to lend their tefillin to others at the minyan wait at least until after Shemoneh Esrei, and usually until after Kedusha.
You ask wisely how the borrower should prioritize between davening as much as possible with tefillin and davening as much as possible (including, especially, Shemoneh Esrei) with a minyan. (If feasible on halachic and technical grounds, davening at a later minyan is, of course, the best option.)
There appears to be a contradiction on the matter between sources relating to your question and between the standard practice regarding a related one. Many people who go to a late minyan “solve” the problem of missing sof z’man Kri’at Shema[4] by reciting it before shul. Several poskim deal with the question of whether this is acceptable, since these people usually recite Kri’at Shema without tefillin.[5] Rav Ovadia Yosef[6] justifies the action by citing authorities who maintain that the problem of “false testimony” is not relevant if one will put on tefillin later in the day.[7] Offering further support is the opinion of the Meiri[8] that saying Kri’at Shema without tefillin is a problem only when it is done in a manner that shows disregard for tefillin. The Meiri further maintains that the whole issue is only Rabbinic.[9] Thus, the practical solution of saying Kri’at Shema before shul has a solid basis; saying Kri’at Shema without wearing tefillin is acceptable under appropriate circumstances.
Based on this, we would expect that that if one has to choose between davening Shemoneh Esrei with a minyan but without tefillin, on the one hand, and reciting Kri’at Shema and Shemoneh Esrei with tefillin but without a minyan, on the other hand, that a minyan takes precedence. However, the Magen Avraham[10] writes that wearing tefillin has precedence, and one should wait for them before davening, even if doing so causes him to miss the minyan.
The Magen Avraham’s analysis is reminiscent of the transitive property of inequality. If one enters shul when the congregation is about to start Shemoneh Esrei of Shacharit, the rule is that he does not skip straight to Shemoneh Esrei. This is so because the importance of davening Shemoneh Esrei immediately following the mention of geula[11] (call this G) is greater than that of davening Shemoneh Esrei with a minyan (call this M).[12] In other words, G > M. Nevertheless, if one attains tefillin between geula and Shemoneh Esrei, he should put on the tefillin at that point, despite the resulting break.[13] This shows that the importance of davening with tefillin (call this T) is greater than the proximity of Shemoneh Esrei to geula: T > G. Since T > G and G > M, it follows that T > M. Thus, applying the “transitive property of importance,” it follows that davening Shemoneh Esrei with tefillin must be more important than davening Shemoneh Esrei with a minyan.
The Magen Avraham’s proof is weak because putting on tefillin between geula and Shemoneh Esrei does not necessarily constitute a full halachic break.[14] Additionally, we can use an analysis similar to that of the Magen Avraham to come to the opposite conclusion. One may recite Kri’at Shema without tefillin in order to daven like vatikin (call the latter V).[15] Thus, V > T.) Furthermore, many posit that davening Shemoneh Esrei with a minyan is greater than vatikin:[16] M > V. Since M > V and V > T, it follows that M > T. Accordingly, applying the transitive property results in the conclusion that davening with a minyan is more important than davening with tefillin, contradicting the Magen Avraham!
Thus, there is significant halachic logic to argue that one who will be borrowing tefillin later on should daven along with the congregation and put on the tefillin during chazarat hashatz when he gets them.[17] Nevertheless, it is difficult to rule against the Magen Avraham, whose view is accepted by the Mishna Berura, and one should therefore wait until after getting the tefillin before reciting the crucial parts of tefilla.[18]
Assuming one is following the view of the Magen Avraham, may he at least start some of Shacharit before getting the tefillin, and if yes, at what point in the tefilla should he put them on? It is better to put on the tefillin right after Yishtabach than during the berachot of Kri’at Shema or P’sukei D’Zimra,[19] but even this is not the optimal time.[20] Therefore, considering that he will not be reciting the critical passages of Kri’at Shema and Shemoneh Esrei with a minyan in any case, there are advantages to hold off saying P’sukei D’Zimra and to put on the tefillin before Baruch She’amar. (There are so many timing considerations involved that it is not possible to give one conclusive ruling or recommendation.)