Hi OP, I haven’t read the whole thread, but thought I would post as I have just been through the same.
My DD is 3 and has just started nursery 2 afternoons a week. At home she was more or less accident free.
The first week at nursery she had no accidents. After that she was coming home in her spare clothes every time, and she is only there for up to 4.5 hrs.
About a month in, she had 2 accidents in one afternoon and I, like you, felt that this was a concern because I had asked them to prompt her, given that it was a new environment, and she may be more easily distracted while playing.
It came to light that her key worker finishes at 3.30 every day and that the accidents were always happening after this time. (She should have been assigned someone else given that she is only there afternoons but never mind.)
I asked them to make sure DD was given a verbal handover to another member of staff so that she knew who to go to and they knew to continue prompting her. She hasn’t had another accident at nursery since.
Every child at nursery should have a key worker, so speak your DS’s and ask them personally to prompt your DS at regular intervals. It is not an unreasonable request. Good luck!