If a baby is sick on the 8th day, the brit is pushed off until the baby is healthy. Once the mohel and doctor determine that the baby is healthy, does the brit need to be performed right away (on that day), or can it be scheduled to happen a few days later?
ואם לא מל, חייב כרת. הגה: ובכל יום עוברים בעונשין אלו.
And if he was not circumcised, he receives Karet (Spiritual Excision). Gloss: And he receives this punishment each day. (Shulchan Aruch YD 261:1)
CYLOM for a practical ruling, but it seems from the above that the milah should be scheduled as soon as possible to avoid excessive Karet.
(I'll note that there is a machloket rishonim brought in the Tur there if every day is Karet or just a Bittul Aseh, but either way better get it over with sooner.)
I found here that Rabbi Paysach J. Krohn, Certified Mohel writes that delaying the bris after the baby is healthy for a more convienient time is not appropriate, but rather it should be performed "as soon as possible".
The Noda Bihuda (Tinyana YD 166, quoted in Pitchei Teshuva YD 262:2) was asked about a father whose son's Milah was delayed for health reasons who now wants to delay until Erev Pesach in order to create a Seudat Mitzva for the local firstborns to eat from. He writes back tersely and very forcibly that delaying the Milah at all unnecessarily is a "very disgusting thing" and should not be done.
My son’s bris was delayed over 5 weeks. It was summertime. The mohel came one afternoon maybe 2 hours before shki’a and said we could do the bris that day. Most of the family were living 200 miles away and could not have attended.
I asked my Rav (who is widely respected in the Community) who said that we should go ahead that day as long as there was no chance of any quarrels being generated in the family by not giving them time to attend. We phoned the family and were reassured and so went ahead.
A friend of mine had a delayed bris for his son. He said the psak he got was that if it is delayed because the baby is yellow, then you do it as soon as possible after the jaundice has cleared. If it was because the baby was sick with something else, you wait a week before doing the Bris to make sure the baby is healthy. See here where it references the same concept, but is more vague about what constitutes a health risk that requires waiting seven days.
In the case of my son, whose Brit Milah was delayed because of neonatal jaundice, the Psak our Rav gave us was to wait until the bilirubin levels come all the way down to the "LeChatchila" level recommended by the doctors, and not just to the "mildly-high-but-not-overly-dangerous" levels at which doctors will allow an 8-day-old baby to undergo a Brit Milah on time.
If I remember the numbers correctly, the "LeChatchila" serum bilirubin levels are 10 mg/dL or less, although at 12 mg/dL it is still possible to perform a Brit Milah. My son had over 16 mg/dL on his 8th day, and our Rav told us that -- since we already going to be late -- we should wait until the bilirubin levels come all the way down to 10, and not just to 12.
The halacha is that for a child that was yellow or had only a localized, mild health issue, we do the bris as soon as feasible after he is declared healthy enough to have the bris by the doctor and mohel.
However, if a child had a fever or some other serious illness, we wait a week (7 days of 24 hours) before conducting the bris. See Shulchan Aruch, Yoreh Deah 262:2.
This halacha is brought down in the Gemara in Shabbos 137a.