We will first explain why this question does not arise in the writings of the classical poskim. A brit mila may be performed throughout the daylight period. However, since “the diligent do mitzvot early,”[1] it is proper that it take place in the morning, and this has been the widespread minhag for centuries.[2] There is therefore little discussion of mila at Mincha time.
Considering why Shacharit precedes a brit may be instructive in attempting to determine whether Mincha should similarly precede a brit. One reason is that tadir kodem (more frequently occurring mitzvot are performed before less frequent ones).[3] This reason would seem to apply to Mincha as well. Alternatively, perhaps the precedence of Shacharit is due to its association with other mitzvot such as Kri’at Shema, which is a Torah commandment in addition to being frequent.[4] This reasoning would not apply to Mincha. Other explanations include that we are concerned that if we delay Shacharit, we may not be able to daven by the end of its proper time, and that Shacharit is a morning mitzva, whereas mila is a daylight mitzva.[5] These latter reasons also do not apply to Mincha, the latest time for which is the same as for a brit.
Is the factor of tadir sufficient to compel us to daven Mincha before a brit? In a similar situation, the gemara[6] says that one who had not davened Musaf by Mincha time should daven Mincha first because of tadir. However, Tosafot[7] claims that this is so only when one needs to daven Musaf and Mincha in immediate succession. Otherwise, one should daven Musaf first and daven Mincha later within its time span. According to this, the fact that Mincha is more tadir than a brit would only give it precedence if it is necessary to daven Mincha around the time of the brit.
The Shulchan Aruch[8] seems to prefer the opinion that Mincha precedes Musaf even when one could daven Mincha later. But how far do we take the precedence of tadir kodem? Certainly, one who wants to recite Shehecheyanu on a fruit (a non-tadir mitzva) in the afternoon does not have to daven Mincha first! Rather, something must trigger the tadir mitzva to “come to mind” now. In the gemara’s case, the fact that one wants to daven naturally raises the question of which tefilla is appropriate. Similarly, when one wakes up on the day of the brit, both Shacharit and mila are matters he ordinarily wants to take care of as soon as possible. In contrast, Mincha is not something that is normally davened at the earliest opportunity. That being the case, the fact that it is tadir may not give it precedence over the brit.
One reason to require Mincha first is the halachic problem of having a “big meal,” including a brit meal, prior to Mincha because of the concern that one will become distracted and fail to daven Mincha in its proper time.[9] However, that issue may not apply to everyone, perhaps including those who are in a situation in which they will be reminded to daven.[10] Furthermore, there sometimes is a long break between the brit and the festive meal, during which time Mincha can be davened.
It appears that different courses of action are called for in different cases. When the “schedule” of the brit, as determined by the principals to the brit, includes Mincha, it makes sense to daven Mincha first because it is tadir, and this is in fact what usually happens. If Mincha is arranged independently by the guests, there should be no halachic question. Neither the mitzva of mila nor the mitzva to take part in the meal (which, as we noted, can generate the obligation to daven Mincha first) are obligations that are incumbent upon guests. Therefore, the guests do not have to decide whether Mincha or mila has precedence. It is a matter of their convenience. Furthermore, there are other factors – such as the mohel’s schedule, among other matters – that may be part of the equation. Thus, no rule or ruling in this regard can be applied definitively across the board.