Many parents of children with SN / development difficulties choose not to tell anyone until we are more 'comfortable' with the condition and situation. My DS is going through assessment, and we have told my parents, but not my sister, with whom I am close, and hardly anyone else. It can take ages (years) to get a diagnosis, if you ever do. It takes at least 6mo to get therapy (for us, raising concern to initial SALT took 5 mo, and that was quick). Similar process times for assessment.
From what you say, it sounds as if there is more that you're not being told, and I think it's quite reasonable for your SIL to choose not to tell you and your DH something that (a) quite understandably upsets her, and (b) she doesn't yet know what the outcome is.
If you and / or your MIL want to help him, then be kind and supportive family, there's probably notheing specific you could do at this stage, that would be different to normal supportive family relationships. Be understanding - I've had times of not enough sleep, too much to do, distress and stress, added to the difficulty of having a challenging child and a baby, and the result is that I've been unpleasant, annoying, snappy, unreasonable etc So I would really appreciate people cutting me some slack and continuing to be friendly even if my behaviour left soemthingt o be desired.
Depending on your relationship and how it would be taken, you could take your nephew out for the afternoon with your DD / offer to mind her baby / just be an 'active auntie'. Build a supportive and non-judgemental relationship from there.
Don't let your 'he needs more attention' attitude comes across (even if that's only partially what you think).
Just what I think, anyway.