Just off entrance to Persimmon Path, growing on an old tree root. The fourth photo most resembles the nickname. Thanks to cmtilden for spotting. My first sight of this species.
At base of European Beech with Xylaria polymorpha observed here https://www.inaturalist.org/observations/120082937
Spotted by cmtilden. Even knowing where to find them I almost missed them. On the packed earth and wood chips in front of monument to Irish soldiers. Here is a key to Cyathus:
https://www.mushroomexpert.com/birdsnests.html