Port Lympne was the only public collection to hold hyena through the 1980s, with striped and brown hyenas in the bottom carnivore enclosures by the lions. I don't think they ever bred, though I could be wrong, and they were sent to retire in South Africa in the 1990s.
Colchester got a pair of Spotted hyenas in the mid to late 80's I think, they still have one of this pair according to ISIS, it must be very old now. Africa alive acquired three striped hyenas in the early to mid 1990s, these successfully raised a litter of cubs, one or possibly two went to colchester who now hold 1.1 as well as the single spotted. Now Africa Alive also just have 1.1, I don't know if these are the two parents of the litter they raised or not.
Throughout this time, the clubb-chipperfield organisation owned striped hyenas, which bred at least once in the 1980's. I assumed they were hand-reared as they made a couple of tv appearances, as these were supposedly the first of their species to be successfully reared in this country. The remainder of this group, or their parents, if there are any still alive, would reside at the Heythop private zoo owned by Jim Clubb in oxfordshire, as part of his 'amazing animals' company, used in TV/film. It is partly for this reason that the striped species has been seen in television adverts and in british films like harry potter over the last decade or so, despite being less common in captivity that the spotted. These are quite possibly the Jim Clubb-owned animals he bred.
If a zoo wished to, I think they could acquire at least striped or spotted hyenas from europe, only all three species are only ESB not EEP status, I don't know if there are any EAZA recommendations to increase the numbers of hyenas in europe.
The related aardwolf are an entirely different matter. There are very few in captivity. When the current animals die of old age at Hamerton and Twycross, that may be the end of this species in UK collections.