In the name of Allah, the most Beneficent, the most Merciful.
There is nothing mentioned in the ahadith of the Prophet of Allah Sallallahu Alahi Wasalam to recite a particular dua before the cutting of the umbilical cord. You can recite Bismillah if you wish. However, it should be understood that one should bury the umbilical cord and not let the hospital take it as they will burn the remains and one is not allowed to burn a part of a human being.
It is similar to a situation where a foetus has not developed any human organs; it will not be regarded as a child. There is no ghusl, no kafan and no masnoon burial. It shall be wrapped in a cloth and buried. (Raddul Muhtar p.131 v.3)
Only Allah Knows Best
Mohammed Tosir Miah
Darul Ifta Birmingham