I can't claim to be any sort of expert, but my understanding is:-
Initially the family said they would apply to the EHCR: that would be an application based on the Human Rights Convention. Then they said they weren't going to do that because the outcome was inevitable, as the EHCR has already considered the issues in question in the Gard and Evans cases and wouldn't find anything different, so they approached the UN instead.
The UN application is based on alleged breach of the Convention on the Rights of Persons of Disability, and possibly the one on the Rights of the Child. They don't have any jurisdiction within our system because, although we've signed those conventions, we've never adopted them into our law as we have with the HR convention. This was raised in the original appeal to the Court of Appeal, who basically said no, doesn't apply here. The UN asked for more time to consider the application, the Department of Health passed the buck to the courts who basically said no.
So the idea of applying to the EHCR came back into play, but as the lawyers more or less conceded that there is little the ECHR can do it's blatantly a time wasting tactic.