Q: Can the battalions in this book be used in Khorne, Nurgle,
Slaanesh or Tzeentch armies?
A: Yes, but the battalion and all of the units in it would be allies
Any warscroll, including battalions, with the nurgle keyword should be able to be taken in a nurgle army, not just including only units with the keyword. (Don't have the book, so not sure if the battalion has the NURGLE keyword) If it has, technically this should not have been an
FAQ but an errata (removing the nurgle keyword from the warscroll).
Otherwise, following the same logic, you could start banning
STD units in a nurgle army or say they need to be allies.