As you are aware, the time to do the brit mila is on the eighth day, but it is not always possible to do it then. After that time, there are several reasons to do the brit as soon as possible.
A simple reason to do the brit promptly is the rule z’rizin makdimin l’mitzvot (the diligent do mitzvot promptly).[1] Promptness is a relative term, and its application depends on the relative gain versus the loss from any delay. For example, if one has a choice between using a mohel of questionable validity on the eighth day and waiting a few days for a proper one, one waits.[2]
The question regarding a delayed brit is in determining what constitutes a significant delay. A regular brit has a deadline of sunset of the eighth day.[3] Here, by contrast, there is no specific day, as the mitzva needs to be done some time during one’s life, but the sooner the better. The Noda B’Yehuda[4] was asked whether it was permissible to postpone a brit until the upcoming Erev Pesach so that the se’uda of the brit could be used to excuse firstborns from their fast, and he vehemently opposed the idea.
One should realize that it is more problematic to delay a brit than to delay other mitzvot. Usually, positive mitzvot are obligatory opportunities to do something positive, and as long as it is done, it is done. However, brit mila, in addition to being a positive mitzva, removes the status of arel.[5] That status has major spiritual and halachic negative ramifications, and should be removed as soon as possible. For this reason, the Magen Avraham[6] allows the mila and se’uda of a delayed brit to take place on Erev Shabbat, whereas one may not do the same for a delayed pidyon haben.
The D’var Avraham[7] went so far as to say that a mohel who must choose whether to first do an eighth-day brit or a delayed one should choose the delayed one. This is because the latter baby needs to have his foreskin removed immediately, whereas the eight-day old has until the end of the day. Few poskim agree with this urgency to shorten the delay by just a few minutes, as long as the brit is in the process of being addressed.[8] Nevertheless, there is a consensus that the delayed brit should be performed on the first possible day, barring unusually difficult circumstances.[9] One should not delay a brit in order to facilitate its being more lavish or better-attended.
Although it is proper to have the se’udat mitzva on the day of the brit,[10] it is preferable to schedule it on some day after the brit, if need be, rather than delay the brit.[11] Be aware that a modest meal with a minyan fulfills the requirements for a se’udat mitzva.