Thanks to John, Jeff and Leroy for your comments. Really appreciated!
I think I will change both the pump and the thermostat. I wasn't planning
on changing the pump but I guess whilst most of the engine is in pieces I
may as well!
Leroy, with regards to your question, the bottom of the rad is clean.
Well, it isn't showing any signs of leaking. I have had it suggested to me
by a friend that the sudden jump from normal to hot on the temperature
gauge could be an indication of an air blockage in the system. This would
fit in with the fact that one of the hoses (exiting the rad) felt empty
(no water inside) and cold!