I suspect that the spouse is dead, and that you will find evidence of this fairly quickly. Otherwise, the companion romance options make very little sense. The Main Character is designed to be (at least for Canon purposes) in love/faithful to their spouse - otherwise, why would they wear their wedding ring? It's not like wearing it is gonna make them more desirable in post-apocalyptia. The child, I believe, will survive, in some fashion. I have a couple of theories:
1) The Vault is "full". In the intro video, the Main Character asks if there is enough room for his family and is assured that there is. This may (and I say *may*) indicate that there really isn't. They get there and only the MC is let in (he's a Veteran, after all). The family will be "transported to a different nearby vault". Or so the MC is told. Perhaps he finds the van they were in abandoned on the road near his Vault with no sign of his family. This would lead him to speculate that they may still be alive, once he sees that Codsworth survived. What actually happened? Who knows? Perhaps they were placed in Cryo as some suggest.
2) The radiation blast "kills" the spouse and child - or so it appears. The MC is rushed down into the Vault and placed in Cryo to be treated for severe radiation poisoning. Meanwhile, the spouse may, or may not be dead, but the child lives on - as a Ghoul.
Of course, I could be completely wrong. But somehow the MC needs to find evidence that at least his spouse did not survive for the romance arcs to make much sense. A spouse who continues to wear a wedding ring, either wears it in memory of a loved one, and removes it once they finally move on with someone else, or they wear it in hopes of finding their missing spouse. If the MC does believe others (his family) could have survived, then something has to remove hope of finding his spouse for him to be able to move on in a realistic way (in my opinion).
I do think that the generation of the child is more effort than what is necessary if it is just a "throw-away" set-up piece. Even in Character Creation, creating the spouse at least gets an opposite-gender character sculpted so that, should you decide to play the other gender (and perhaps they have data that suggests enough people like to do that to put in the ability to do so), your other character is ready to go. But the baby just doesn't make sense to me. Now, whether the child is a villain, a future companion, or something else, I am not sure. But it will greatly surprise me if the child is not used in any other way besides to be killed, along with the spouse.
(I have not read all of the previous posts, so forgive me if I have re-posted what others have already said).