This has already been done for the Netherlands a few times, and is repeated every 5 or so years. See all the data in your browser here: http://ahn2.pointclouds.nl/
Not sure if this is the same map you're talking about, but Ordinance Survey made a map[0] using data they had in 2013, and then again in 2014. This was based on height data at 50m resolution.
It's hard to see how OS capture height exactly, I imagine via aircraft[1]. Not sure whether they are the source of the existing data the Environment agency has; I doubt it as their data has a max resolution of 5m while the government's is 1m.
Most of the OS terrain data is now from pairs of aerial photos using photogrammetry. The older products were actually based on contours that were surveyed using more traditional techniques.
Topographic (geodetic) shape is always changing with time.
Radar inferometry from satellites gives the most repeated surveys, but has low spatial resolution.
You can discern made-made, seasonal, and geological factors causing change.
The American Geophysical Union meeting each December has several hundred papers on dynamic geodesy.
OpenStreetMap does not have much of its own elevation data. (There is a way to tag elevation, but it is rather seldom done). Any project that shows you OSM data and elevation is probably pulling in third-party elevation data such as that from the Shuttle Radar Topography Mission (SRTM)[1].
I was pleasantly surprised to learn that Airbus sells a very nice Digital Elevation Model. It seems to be quite a lot better than the shuttle data everyone was using for a while.
UK Lidar: https://data.gov.uk/dataset/lidar-composite-dsm-1m1
There's even some Unity plugins that will merge this lidar with satellite photos and openstreetmap and get you a 3d environment from all this data.