Is it possible to extract the depth of an inland lake from a DTM or DSM? Is it recorded or just the surface elevation of the water?
It depends in the method in which the DTM/DSM were derived, and what data they are based on.
LiDAR, which is often used today, sometimes cannot penetrate water, therefore, datasets based on LiDAR typically do not contains bathymetry.
There may be some hybrid products available where the above water surfaces are LiDAR derived, while the under water surfaces are produced in some other way, and then combined.
In my experience, you will mostly see one or the other.
If you look at your datasets' metadata, you will find that the UK Environment Agency's LiDAR-derived DTM is, obviously, LiDAR based, and therefore does not contain bathymetry, and the OS Terrain 5 DTM states that it represents lakes as a flat surface (page 15)
And thanks to @JeffreyEvans for pointing out the following in the comments:
With lidar it depends on the sensor. The blue-green sensors, currently being flown by many vendors, bathymetry is included with the terrestrial data. With NIR sensors, water causes scattering so, water is a null in the point cloud being filtered out during pre-processing. When the point cloud is interpolated, to generate a DTM/DEM, water is represented as a flat area based on the surrounding data where the returns drop out.