The Ordnance Survey doesn't even hold height data on all buildings in the UK. They've started a few isolated projects but nothing as a deliverable dataset yet. I don't think there's a US equivalent despite your link to NYC Open Data.
You might want to look to Germany and CityGML. You might discover some town datasets that are complete in height and geometry detail.
If you happen on a LIDAR data source you've then got to tidy up against the buildings but you will run into errors caused by the time spent between the two data capture points. There's SRTM but i'm sure it'll be too coarse for your needs. I'm sure there's a replacement to SRTM but the name escapes me.
There's LandXplorer - now owned by Autodesk which will generate models from oblique aerial photogrammetry. There's also CityEngine that'll create fake cities based upon road networks.