Hello OP
I am sorry for your loss 
There isn't a UK-wide rule for live-streaming funeral services; local councils have interpreted the guidelines differently. Have you checked the specific circumstances for the area where the funeral was held?
I've 'attended' three funerals and one wedding during lockdown. For the funeral services (two church, one crematorium) the camera was in a fixed position and showed a single shot of the lecturn/pulpit. The edge of the coffin could just be seen. Only people who came up to read were on camera.
I would assume the set-up was similar, i.e the crematorium had a small, fixed camera? You wouldn't see the camera in my own church even if you knew where to look, so I understand why it wasn't spotted/queried on the day.
The funeral director should be able to tell you exactly what was broadcast. Hopefully they will be able to reassure you regarding your children being on the footage.
Again the guidelines may have been interpreted differently, but I believe that all funeral service live-streams must be protected by a username and password. It's given to the person who paid for the funeral, meaning it's their choice whether or not to share it.
If the live-stream was username/password protected, the company will have records showing how many times it was accessed and from which IP addresses. You won't be able to find out which individual people they belong to, but you'd know how many people watched the service.
I think it would be best if you asked the funeral director to make these enquiries to the crematorium on your behalf (who gave permission for the live-stream or is it a local council rule that all funerals are streamed, who was given the username/password etc.)
I don't believe there are grounds for a legal claim, however the crematorium should definitely investigate and apologise if the guidelines/rules have not been followed correctly.
One final thought - are you SURE the ex saw a live-stream of the funeral and isn't trying to wind you up? This is exactly the sort of thing my abusive ex would have done.