All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The majority of the scholars are of the view that the 'Aqeeqah is a Sunnah (recommended act) and not a religious obligation. It appears that slaughtering the animal on the seventh day for the 'Aqeeqah (i.e. the slaughtering done for a newborn) is more rewarding than slaughtering it on the 14th or 21st day after the birth of the child, for whoever is able to slaughter, because slaughtering the animal on the seventh day is what is required, and by doing so, one hastens to act according to the Sunnah.
Al-Mardaawi from the Hanbali School of jurisprudence, said: "Slaughtering the 'Aqeeqah on the seventh day is better."
However, it is hoped that whoever delayed it for a valid reason will get the full reward, by Allaah's Grace and Mercy.
As regards your saying that women after giving birth to the child cannot wear new clothes before 40 days elapse, then there is no evidence from the Book of Allaah or the Sunnah of the Prophet about this. So she can wear new clothes especially if she wants to show her happiness about the Blessing of Allaah [on her]. Moreover, the Prophet said: "Indeed, Allaah likes to see the effect of His Blessing on His Slave." [At-Tirmithi]
Allaah Knows best.