Great argument developing here gents. If it gets any more convoluted it's in danger of disappearing up its own bottom.
However, I suggest the arguments about NATS' responsibilities to the SVFR community and how this may or not fit into the brave new era of part-privatisation are irrelevant.
CAP493 (MATS Pt 1 to you and I) - CAA SRG document and therefore not the doctrine of NATS regardless of how it is funded, states (Section 1, Chapter 2, Page 5, Paragraph 8.1, line 4, and commencing 9 words from the left-hand side):
"Special VFR flights are not to hinder normal IFR flights"
So, if available human resources cannot support the full Thames Radar/SVFR manpower requirement (for whatever reason - and I suggest unusual levels of sickness is a perfectly valid reason), then SVFR will be the one to go - not because of who is paying for the service - but because the CAA says so.