Babies who are born outside the UK to parents who have long-term visit visas need to make an application that is fundamentally no different than any other applicant.
The form should be filled out by the primary and the primary should include a notarised scan of their entry clearance and the put their own GWF number in the remarks section (if available). If you plan for both parents or either parent to travel alone with the child, then both parents should submit notarised scans and their GWF numbers.
The big difference for this type of application is that (in addition to the Immigration Rules) Section 55 of the Borders Citizenship and Immigration Act 2009 kick in and it's something they have been taking very seriously. Parliament has also told them to help enforce the various provisions of the Children Act 1989 concerning ALL children in the UK (it means they can and will challenge even Brits coming through if they want).
So evidence that establishes true distaff and spear side parentage must be included. They have the option, but not the obligation to get back to you during the application stage to get DNA testing done. You should also provide some very detailed evidence about child care for the times the child is in the UK and if anyone other than the parent will be providing care, information about them. I cannot stress enough UKVI's sobriety and determination where children are concerned.
You asked about what length of visa to ask for. Compare these three dates and select the earliest: expiry of the child's passport, expiry of the parent's passport, expiry of any parent's entry clearance. That's the length of visa you should ask for. They will not issue a visa to a child where the length extends beyond the accompanying guardian(s),
Finally, since having a baby is a big change of circumstances for YOU, and hence reportable against the terms and conditions of your own entry clearances. You would do that at your next landing interview. To play it strictly by the book you can say "I have a change of circumstances to report." They will know what to do after that. It's not a big deal as long as all the entry clearances are in order (you will get a +1 in your transcript) and it gets you off the hook in future applications for not reporting it.