Hi Poppet, have a look at Part 3 of the attached especially point 34 on p.24
https://www.gov.uk/guidance/party-wall- ... 6-guidance
I suspect they are incorrect about not needing planning permission as even though the return is less than 3metres wide it'll probably be over 3metres long, this means it doesn't fall within permitted development and requires planning permission. Also I don't really understand how they are avoiding the need for a steel? As soon as the current side wall is removed then the house will be unsupported.
Have you spoken to the people living in the upstairs flat to where the works are being done? As this work impacts on them too and they also should have party wall notice, statement of condition etc..
Good luck, what an anxious position to be put in.