Apparently these fellowships are popular in the UK? What sorts of universities care about them? And why?
(Nearly) all UK universities want to signal that they take teaching seriously, both for student recruitment and for the TEF, for the obvious financial reasons.
For this reason new faculty are typically required to undergo training to reach a qualification that either leads to an HEA fellowship or is equivalent to it. Thus HEA fellowships are the national standard for teaching. Typically new faculty are exempt from these training courses if they have an HEA fellowship.
Anecdotally various people within my own institution have said that the push to get more faculty accredited with HEA fellowships was important for an improvement in our TEF ranking. There continues to be a big push to get more existing faculty to do the paperwork for HEA fellowship, although I do not believe it is a requirement (or even mentioned as desirable) for job applications for the majority of posts.
From my very limited experience, I would say universities that participate in TEF would care about having academic staff with HEA fellowships (and there are many levels of fellowship). In my view, getting one will only work in your advantage. For what it is worth, I have been asked about having/willing to get the accreditation during job interviews, so I assume the universities (Russel Group) definitely care. I've also been told that having/on track to getting one will look good for annual review (it counts toward part of your personal development). Finally, getting a fellowship (above the Associate Fellow level) entitles you to add FHEA after your name. I personally find the process very rewarding, as it really forces you to reflect critically on how you teach (not just what you teach). As most PhD programmes don't teach you how to teach, going through the trouble of getting a fellowship provides that opportunity to be trained.