May a relative or friend accompany the Yoledes to the hospital?
Any relative or friend who will help the Yoledes keep calm may travel with her. This applies even if there are other people in the car, such as a medic or EMT, and even if the Yoledes says that he/she does not need the escort of the relative or friend. [However care must be taken to avoid Chilul Shabbos in the process, such as not to carry items with him/her if there is no Eiruv.] However, there are Poskim who are less lenient and only allow an escort in a case that the Yoledes is asking for one. Others allow it in the following cases: a) A first birth even if she says she does not need an escort. b) A 2nd birth and onwards so long as she does not say that she does not need it. c) Even if she says she does not need it by a 2nd birth and on, but she will be left in the ambulance in the care of gentiles. Thus, only if it is her 2nd or subsequent birth, and there is a Jewish driver or Medic with her, and she says that she does not need an escort should a friend or relative refrain from escorting her.
May more than one person accompany the Yoledes to the hospital, such as her mother and husband?
Seemingly this would depend on how badly the Yoledes wants their company, and for what reason, and if any Shabbos desecration will take place by the person accompanying the Yoledes [i.e. gentile driver within Techum Shabbos etc]. If no Shabbos desecration will take place, then it is permitted to be done.
Having a doula accompany the Yoeldes to the hospital:
It is permitted to have a doula accompany the Yoeldes to the hospital if the matter is very pressing for the Yoeldes, as she will feel anxious giving birth without her help and support. One may even desecrate Shabbos for this purpose and may thus call the Doula on Shabbos in order to inform her to come. If, however, the Yoledes does not see the service of the doula as necessary for her calmness and serenity, then she should not desecrate Shabbos in a biblical way in order to come to the hospital. However, if going to the hospital only involves rabbinical prohibitions then she may go.
If on Shabbos one traveled to a hospital outside of Techum Shabbos as an escort for a Yoledes, may he/she walk out of the hospital?
Yes. One may walk within 2000 cubits from his destination. If he arrived in a city, he has the same Techum as do the city inhabitants.____________________________________________
 See Ketzos Hashulchan 140:3; Igros Moshe 1:132; 141; Piskeiy Teshuvos 330:1 footnote 3; Nitei Gavriel 58:8-10; Toras Hayoledes chapter 14
 Koveitz Igros Chazon Ish 141; Az Nidbaru 1:29; Beir Moshe 6 Kuntrus Elektry 51; Nachalas Meir 16; Toras Hayoledes p. 80
 So is implied from Igros Moshe 1:132
 Teshuvos Vehanhagos 2:177
 See Nitei Gavriel 58:9; Piskeiy Teshuvos 330:1
 Kinyan Torah 4:38; Igros Moshe ibid
 See Toras Hayoledes 35:26; Piskeiy Teshuvos 330:1 footnote 5
 Based on 407:1; See Toras Hayoledes Chapter 29
 As this is similar to a midwife which has traveled to help a woman give birth. Likewise, the escorts are also considered to have traveled due to Pikuach Nefesh. Now although in 405:5 Admur rules that “when one left the Techum due to Ones then if there is a Kosher Eiruv in the city one may walk anywhere in the city. If there isn’t a Kosher Eiruv in the city, then one may not leave the hospital building for any purpose.” Nevertheless, in a case that one left to help save a life the Sages did not limit them to their 4 cubits even if there is no Eiruv in the city.