Ugh, what a terrible position to be in.
I would wait to apply until July 2nd if need be, so you can have the LIUK test done (if it's available). If by that point you still haven't been able to sit the test, THEN include all the cancellations, etc. with a cover letter explaining. It may be that they grant you a short extension to get it done, etc. We will know more by then.
There is no point in trying to get the biometrics done before baby unless they reopen the biometrics centres (all currently closed). And honestly, biometrics will baby isn't an issue. They are mobile (babies) and they take only a couple of minutes.
I took my LIUK test 10 days before my second. My biggest worry was needing to pee. The woman running the centre understood my issue and let me pee immediately before and immediately after the test.
But thankfully I took the test about 6 months before my ILR was due. But obviously no one could have foreseen the country literally closing until about a month ago.
Try not to worry. There *will* be an update on guidance about all of this in due course.