it appears the configurations that NATS uses include everything
No, it doesn't. As it happens, I was including ANOMS in my reference to "airport NTK systems that I've encountered".
If you look again at that diagram, the part we're interested in for track monitoring is the box labelled "t,x,y,h" (i.e. aircraft's 3D position plus time). Follow that link upstream and it ends at the box labelled "Radar". No ADS-B.
I agree with your point about WebTrak missing flights, though, in fact a couple of years ago the LHR system went through a phase where it used to go sick at weekends and lose dozens of flights. On that occasion I got involved in helping out by providing independently-captured ADS-B data to assist BAA/NATS to troubleshoot the problem.
It still drops flights from time to time, though.