Step 1 is to wait for the British Passport to arrive. This can take anywhere from 2 to 6+ months and there doesn't seem to be any "magic" set of documents or requirements that you can fulfil in order to change that timeline.
In case you haven done so, you need to also email your local FRRO office (
https://indianfrro.gov.in/eservices/) to notify them of the birth of your child within 14 days of birth. Otherwise, you will likely be charged a penalty when you apply for a visa. Assuming based on the date of your initial post that you've applied for the passport last month, in which case you've already passed the deadline. I don't know if it changes anything, but I'd suggest you still go ahead and notify the FRRO.
Once you receive the passport, you need to apply for a New Visa (Category - Entry Visa, Sub Category X1) for the child via the FRRO eServices website (
https://indianfrro.gov.in/eservices/). This visa will take 10-15 days to arrive (via email) and will allow you to continue staying with the child in India for a certain period or fly out of the country.
You can only apply for an OCI from India if you are normally resident in India (>180 days in a year).
