The "Notice of Industrial Action" must contain certain information otherwise it can be challenged in court - see
here (section 3A). You would think this stuff is basic but you'd be surprised what employers would seek to challenge. Specifically it must state how many people are affected, where they work, what "category of employee" they are (to help the employer plan around any action), etc.
The original court challenge by BA related to the "Notice of Ballot" (not of Action) I think, which has a very similar list of requirements. I believe BA's challenge was basically that BALPA hadn't categorised the pilots into SH and LH and so had not met the criterion to provide a list of "categories of employee". The Court disagreed with BA and found BALPA had indeed provided the required information acceptably.
In short, yes BA can challenge the Notice of Action in court, but I think it's unlikely given that they've already tried that with the Notice of Ballot and BALPA will be wise to it.
As others have said, no need to make any information public, but it's still a bit weird - most unions once they deliver that notice would want to shout about it, at least to their own members!
EDIT: I don;t actually know if a Notice of Action has been submitted yet - someone earlier in the thread wondered whether a court challenge was still possible by BA and above is an outline of how action could come about (unlikely)