The application for permission to appeal will almost certainly be on points of law (probably the grounds on which they sought permission to appeal previously). It’s very rare to obtain permission to adduce new evidence (and permission to do so won’t be granted unless it is somehow new evidence that couldn’t have been available at the first stage).
Keira’s team can put in a statement in response on why permission to appeal should be refused but won’t normally be allowed their costs of doing so.
Determination of permission to appeal applications on paper can take anywhere between a month and several months, although there might be grounds for this to be expedited given the arguable risk of harm to children (either way). I wouldn’t expect a decision on whether they have permission until around the end of January at the earliest, but I have known permission to be granted sooner.
If they obtain permission, a hearing of the appeal will be listed. If it’s expedited, this could take place within a few months. In the normal course, it could take a year or more.