i hate to be a show stopper here, but on a literal reading of 319J(e)(iv), the child could only receive ILR if they applied at the same time with the father (in this case)
now, it would be required for the mother to also be applying
it is though possible to make a case that aplication within 28 days is akin to a simultaneous application in all respects, but you need to reiterate that, perhaps without pointing out the relevant paragraph specifically
but if you leave it longer, the kid will be stuck and not eligible for pretty much anything until both parents are settled
plus i have always held and continue to hold a view that pre-july 2012 PBS dependant loses status once the principal migrant gets ILR, which was the position prior to 2012 reforms, and that the 2016 validity does not apply to your wife. dependants' levae used to disappear once the principal migrant changed status, and there is no reason to suggest it is now any different for people who became dependants under pre-july 2012 rules
i know HO caseworkers are sometime too dumb to see it, and this all works -- but i'd not be comfortable with that -- i'd advise her to apply for at least something within 28 days of your ilr
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