Hi Paul, I had a SIII Daimler Sovereign with the same XK engine and suffered the same problems.
Possible causes:
1. Water pump not pumping; see if belt slips, check pulley and belt, smears are telltale, eventually a toothed belt will slip a lot less than a smooth belt
2. Water not flowing evenly and freely within the engine block - gunk? deposits over the years? You need to "wash&rinse" the waterways (*)
3. Thermostat not opening properly; replace thermostat
4. Fan turning too slowly or has too few blades (It's warmer these times compared to 50 years ago); try an 8-blade fan from the S3 Sovereigns, and/or a "blowing" electric fan (Kenlowe etc.) on the front side of the radiator (very thin and will not show through the grille)
(*) You can also consider a "waterless" cooland liquid, which will work from -40°C to +140°C; this means no issues with antifreeze in the winter and safe operating temperatures up to and beyond "off the scale" readings of your temp gauge - it's worth considering...
Click here to read about waterless coolant
Cheers,
Maurizio, Rome, Italy
[note: same post as in your earlier thread, but chances are you've abandoned that and concentrating here]