Alex, I looked through the TFL APIs, documentation and datasets. The station list (KML) format indicates it has locations for light rail (DLR), tube and overground. But as you observed there are no entries for overground stations. I also did not find anything on National Rail in the documentation.
I did find some additional resources. Below is the National Rail's list of stations in London (which you would need to scrape):
The Guardian maintains its own downloadable dataset of all rail stations in Great Britian. .
It appears the Guardian reversed engineered the dataset from the Office of Rail Regulation' station usage statistics.
The UK data.gov has a dataset labeled: National Public Transport Access nodes (NAPTAN).
NaPTAN is a GB national system for uniquely identifying all the points
of access to public transport in GB. It is a core component of the GB
national transport information infrastructure and is used by a number
of other UK standards and information systems. Every GB station, coach
terminus, airport, ferry terminal, bus stop, etc., is allocated at
least one identifier.