On my last visit to the Welsh Mountain zoo (in May 2018) there was still a free minibus service from Colwyn Bay train station(hell of a long walk without it). Chester on the other hand haven't had a minibus service for a number of years but anyone that doesn't drive would have to take a 40 minute roughly bus journey and I had to get the train as well back in 2010 so you may have to travel to the city to get a connecting bus which makes a long day.
I wasn't aware of Chester having ever had a (dedicated) minibus link - I was thinking of the zoo-sponsored service that used full-sized buses. More recently it was linked to the station by the park & ride then by the Stagecoach Liverpool service which is timed to take roughly 15 minutes from the railway station and 20 minutes from the bus station.