Geoff Miller wrote:

If you look closely, this is a J-model. Note the barely
visible dark patch at the bottom of the vertical stabil-
izer's leading edge. (What's that for, anyway? Is it a
dielectric covering for some sort of antenna? In any case,
only J-models have it.)

It somewhat resembles the icing indicators painted onto the wingtip leading edges of all C-130s, but I seriouisly doubt it has anything to do with ice indication. The wingtip indicators are adequate, plus I don't think it's possible to see the one at the base of the vertical stab from inside the aircraft.

Somebody on some website called it a di-electric panel, just as you seem to think. That person wondered if it houses the HF antennas since J-models don't have HF long-wire antennas. I'm not sure that would be the case either though since USAF C-130H3 models also lack HF long-wires and they do NOT have that black panel.

I'd have to wonder if it covers some NVG lights or something.

See how the lower fuselage kicks up immediately aft of
the paratroop door? This is a standard-length C-130,
not one of the stretched versions. On the stretched
ones, the lower fuselage continues horizontally for a
few feet further aft before reaching the leading edge
of the ramp.

Correct. In the US, we call the stretched version a "Dash 30" version (short for C-130J-30). The J isn't the first model of C-130 to have a stretched (-30) fuselage. Lockheed did make previous versions, but I know the USAF never bought any of them prior to getting C-130J-30s.

Verdict: Since an unpainted American C-130 is unlikely
to be in UK skies, it's almost certainly an RAF Hercules
C5.

If you look closely, it also does have the ARR probe. It's easier to see the tip of it if you zoom in some. Since it clearly has an ARR probe, it is definitely NOT a USAF bird and almost certainly a RAF bird.