I am pleasantly surprised by this. Having done a little reading, it seems that several sources confirm this. I would suggest there may not have been more than 2 or so Kings, and that these would have been birds moved from the old exhibit. This anecdotal account mentions them:
The Penguin Coast at Bristol Zoo
Interestingly, this print suggests this species was actually being bred at Bristol as late as the mid-eighties:
https://www.art247.com/Canvas/13321-Pengy-the-Penguin--King-Penguin-born-at-Bristol-Zoo
This quote also confirms the penguin species that the exhibit opened with:
"Seal and Penguin Coasts is a £2 million exhibit at Bristol Zoo Gardens opened at the end of June 1999, exhibiting 3 species of penguin (African, gentoo and king) with flying birds (eg Inca tern and European cormorant) and South American fur seals."
- Evaluation of visitor reaction to ‘Seal and Penguin Coasts’ at Bristol Zoo Gardens
Stephen P. Woollard, Bristol Zoo Gardens, from BIAZA Research News, Vol 1, 2000.
Either the kings were quite old, or they quickly succumbed to something in the new exhibit, as they didn't last long there. As other members have mentioned, the late 90s were a period of several outbreaks of avian malaria and aspergillosis in UK penguin stocks, and may account for the sudden disappearance of a number of penguins from various UK institutions. This article mentions Bristol losing eight penguins in 1999 (the year Seal and Penguin Coasts opened):
http://news.bbc.co.uk/1/hi/sci/tech/467989.stm
I would be saddened, but not totally surprised, if the remaining rockhopper and macaroni penguins at Bristol were taken out by foxes during the 90s, until only the kings and some African penguins were left to move to Seal and Penguin Coasts, at which point the zoo clearly brought in additional African penguins and a new group of Gentoos for the new exhibit. What I hope happened, is that some kind of exchange occurred where Bristol recieved its gentoos for its Rockhopper and Macaroni penguins. I would expect the gentoo penguins to have come from Edinburgh, and possibly Bristols rockhoppers and macaronis went up there?
Can anyone shed any light on that?