cack-handed good intentions
Agreed. This is really an case of misguided good intentions. They mean well, but they've gone about it in a completely cack-handed way.
Just because the NHS is a government agency, does great work, and is under-funded, doesn't mean they are a charity case who'll eagerly snap up everything that any civic-minded group might do for them. They have to follow processes, just the same as any other massive organisation. Probably even more so.
NHoS are a self-appointed group who decided to appropriate NHS logos for a project that NHS England didn't ask them for. It's not unreasonable to the NHS to then worry that people might think that there is an official project, when they have absolutely no over-sight or control of it. How are they to know what kind of quality control is being applied? How are they to know that it won't turn into an embarrassing mess? Can you imagine the headlines? "Broken NHS system depends on 4 men in a cupboard."
The fact that they'd already re-branded from another name suggests that they really weren't paying attention the first time.
If they had wanted to produce something for the NHS, the way to go about it is to create it first, under a completely unrelated banner, and then impress the NHS how good it is, and it's free! Why couldn't they do that?