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.
Most universities in the UK get the majority of their income from undergraduate and postgraduate taught students. Even in a research intensive university most research and teaching (R&T) staff spend more time on teaching than on research. It therefore pays to be able to signal that you value teaching. One way to do that is to say that all/x% of your staff are qualified up to a certain level. As others have pointed out such statistics may affect a university's TEF score, but universities were trying to get everyone qualified well before TEF.
If you are applying for a R&T job at a research intensive university, already having AFHEA, FHEA or SFHEA is unlikely to be the decisive factor in hiring, although it can't hurt. As @DanielHatton pointed out, almost all universities will require R&T staff to gain FHEA or equivalent within the first 3 years on the job as a condition of passing probation.
However, if you are applying for a teaching focused position at a research intensive university, OR any position at a teaching focused university, being qualified might well improve your chances of getting the job significantly.
I guess even non-UK universities might care about it as a signifier of a serious commitment to teaching if you were applying for a primarily teaching focused position.
At Plymouth University, attaining FHEA is a condition of passing probation for all teaching personnel on a contract of 50% full-time equivalent or more; and attaining AFHEA is a condition of passing probation for all personnel on a contract less than 50% full-time equivalent, but who have at least 15 hours per year of student contact time. Policy here. Incidentally, this policy was already in place before TEF was introduced.