Continually requesting:
https://talks.ox.ac.uk/api/talks/search?from=today&organising_department=oxpoints:23232565&organising_department=oxpoints:23232691&organising_department=oxpoints:58125444
gave variable results - sometimes nothing, sometimes some of the talks. I haven't had time to work out if there is a pattern to it (or try it with other departments).
There is a workaround which seems to work so I've set this to 'medium' for now. Add all three departments to a public collection and use the collection endpoint. If using the widget you then need to tweak the URL parameter to accommodate a collection.
Will add more when time allows me to investigate further.