When I first saw The Ferryman in London, the baby did get a bit fussy and cry during the second scene the baby appears in. I'm fairly certain that the actress who plays Shena (Carla Langly in both London and NYC) is the only person who ever holds the baby. She handled it well, but Langly definitely worked a bit to try and calm the baby while still delivering her monologue. I don't remember enough specific details of what happened, but I remember having a sense at the time (which I could have been wrong about) that the actress exited the scene a bit earlier with the fussy baby than she usually did. At any rate, looking over the published script, it says she runs offstage to get her sisters after Aunt Maggie starts talking and then she "reenters (with the baby)," but I do remember that she reentered without the baby at that performance.
When I saw the production again in NYC the baby that night sort of did the opposite and could not stop smiling and laughing at the audience and other actors. The audience got a kick out of it and kept laughing along, which seemed to make the baby laugh more--but luckily it didn't really seem to detract from the play, at least for me. But my companions and I joked after that the baby might be let go for constantly pulling focus!