Can a soldier miss drill a week before their due date? Or, does leave have to begin once labor starts?

Short answer: "Can a Soldier miss drill" - Yes, of course. Just like any other possible excused absence if there is justification.

However, I believe what you're really asking is "when does the Army's policy regarding childbirth come into effect for reservists attending drill". The answer to that is that UTA absences related to pregnancy occurs after the 'conclusion of pregnancy' (birth, miscarriage, etc.)

Prior to the pregnancy conclusion, If there are complications or other issues related to the pregnancy, then they will be handled as normal regarding drill attendance (medical excusal, reschedule for personal conflict, etc).
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* Army Directive 2022-06 (Parenthood, Pregnancy, and Postpartum) - https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN35255-ARMY_DIR_2022-06-000-WEB-1.pdf

That would be best if asked to your chain of command, since they are the approval authourity.

You need to open lines of communication with your chain of Command. They need to be dialed in to this in advance