??? Alfie Evans had nothing to do with the NHS not wanting to cover the treatment. From the first paragraph of the wikipedia article you linked to:
> Alder Hey Children's NHS Foundation Trust sought a declaration that continued ventilatory support was "unkind and inhumane", and not in Evans' best interests
Some people liken that to a "death panel", in that it was not unanimously agreed with by other medical professionals, including those that thought there may be other forms of treatment for the same affliction that were not being covered by the NHS.
Yes, doctors here (as, indeed, in most countries) can take cases to court when they believe the best interests of their patients are not being looked after by the legal guardians. And these are not mediocre doctors, nor was this being done lightly.
The treatments not covered were not even approved for use in the UK privately afaik - it was entirely unrelated to the NHS budgets.
> Alder Hey Children's NHS Foundation Trust sought a declaration that continued ventilatory support was "unkind and inhumane", and not in Evans' best interests