Which Hornbeam are you searching for? With the reference to UK you are probably looking for the european one but there is a native hornbeam also. Several nurseries are supposed to carry the American one including Forest Farm

I was looking for European Hornbeam and/or European Beech. We've grown the native & it isn't quite as... well mannered.
Thanks for the tip- called up Forest Farm & they didn't even have the American version. Still looking but I'm inches away from having a friend ship a few dozen from the UK. Perhaps going through a landscape architect is the key?
This is starting to feel like searching for Heleborus when I was 16, in the mid-nineties when they were still hard to get- I ended up ordering Heleborus SEEDS from the UK.