If a woman goes into labor on Shabbat and must be taken to the hospital, may her husband and mother join her?
The Shulhan Aruch (Orah Haim 330:1) writes that a woman in labor has the status of a "Holeh She’yesh Bo Sakana" – an ill patient in a condition of potential risk to life. As such, the Shabbat prohibitions are waived for anything that is necessary to help her, which includes anything necessary to keep her calm and relaxed. Halacha permits turning on the lights for a woman in labor on Shabbat even if she is blind, and will not directly benefit form the light, if she will be comforted knowing that the room is illuminated. This demonstrates Halacha’s understanding of, and sensitivity to, the emotional wellbeing of a seriously ill patient, and its recognition of the importance of keeping the patient at ease.
Certainly, then, anyone whom the woman wishes to accompany her to the hospital for her comfort and peace of mind may do so.
Preferably, a non-Jew should drive, because then no Torah violation is involved, as asking a non-Jew to perform a Melacha on Shabbat is forbidden only by force of Rabbinic enactment. This is permissible on Shabbat even for an ill patient whose condition is not potentially life-threatening. In the case of a woman in labor, even Torah prohibitions may be violated if necessary, but if it is possible to hire a non-Jewish driver, this would be preferable.
Summary: If a woman goes into labor on Shabbat and must be taken to the hospital, anyone whom she wishes to have with her for her comfort and peace of mind is permitted to accompany her. If possible, a non-Jewish driver should be hired.