There is no published rule that a birth must fall before an interview, so do not plan around one. What the instructions do require is a timely, truthful update when the family composition changes. Raise the expected date with your representative now, so the update route is agreed before the certificate exists.
Planning around a due date you cannot move
Official material describes processing in weeks, which is a description and not a guarantee; a qualified estimate from someone reviewing your file is more useful than either optimism or a flat claim that nothing can be predicted. A scheduled interview does not by itself make the outcome certain. Remember also that a pending case and a paid fee do not authorise work; any work before residence rests on separate, valid authorisation.
The published sequence is application and processing fee first, vetting second, gift third, and immigrant processing under the carrying category last. The framework's own stages are fee, vetting and then payment, and vetting is the stage without a reliable published duration. If an expected transfer is delayed, preserve the confirmation and reassess the timetable before assuming a private payment changes the official sequence.