Mistakenly painted a newly plastered wall with emulsion .

i would say the emulsion you have used has been in part successful. It very much depends on the density of the paint you used example if a quality trade paint was used that would have been a disaster but a cheaper paint wold generally be less dense so you will probably get away with it. It seems that as you have only been able to get off less than half of it it seems to have soaked in and therefore bonded, and should be ok. Yes re-mist coat at around 60/40 paint/water. Hope that helps and good luck.

There is no such thing as a mist coat this is a term used in the early days of emulsion when it took over from distemper or whitewash . The emulsion was not very good at covering so we used to thin it with water and called it a hit n miss coat then we would give the area 2 coats of un thinned emulsion on top. so don't waste your time washing the paint off paint it again, I would not thin down emulsion even on new plaster.