Yes, you're right.
If the child was born in the UK before one parent had ILR/PR, then once one parent is granted ILR, the child is entitled to be registered as British citizen under section 1(3) of the British Nationality Act irrespective of their immigration status.
This application is an entitlement and as such the HO has no discretion in the decision if the statutory requirements are met. The application can be made any time before the child is 18. You just need to submit your child's full UK birth certificate (with your name on it as the father) and evidence that you have obtained ILR. A marriage certificate is required only if your child is born before July 2006.
Below is another link that you may find useful:
http://www.ukba.homeoffice.gov.uk/briti ... /borninuk/