Dear questioner,
Thank you for your question.
A woman in labor is considered a cholah sheyesh bah sakanah (a person with a potentially life-threatening medical condition) and therefore we are allowed to violate any necessary prohibitions to save her life. Thus, driving to the hospital on Shabbat is certainly permissible in the case of pikuach nefesh (saving a life). However, physical actions to coach the woman during labor (such as back-rubbing, etc.) are generally not actions that involve pikuach nefesh and thus would not be permissible for the husband once the wife becomes a niddah during childbirth.
There is a further concept of yishuv hadaat (doing what is necessary to ease her mind) for a woman in labor. Nevertheless, it does not permit the violation of halacha if yishuv hadaat can be reached in other ways. In most cases, the important need for emotional support of the wife can be met by encouraging words. Her need for physical touch can be met by others, such as a nurse or attendant. It is best to arrange for a female labor coach, doula or female relative to provide physical support during childbirth. Therefore, we generally do not allow husbands to touch their wives during childbirth. Passing objects, taking care not to touch, is permissible.
If the wife needs her husband’s physical touch because she is in a dangerous situation (e.g. she needs help to get into a better position for labor and there is no one else who can help her), then the principle of pikuach nefesh would apply. Therefore, if no one else is available, he should help her and not let her suffer or wait unnecessarily. It is best if he does this without direct contact, and touches her only through clothes, while holding a towel, etc. If she is panicky or in great emotional distress, and does not respond to words alone, it may be pikuach nefesh to calm and reassure her physically.
We hope that you will have a positive uplifting childbirth experience, with the emotional support of your husband by your side, even if he cannot physically assist you through labor.
B’Hatzlacha!