I am also leaning towards the water pump though I think that if it is failing the car will overheat more often and not only uphill... I tried to push the car very hard on a flat surface and the needle never moved away from 90 degrees...
Unfortunately your probably not comparing apples with apples here. Engines tend to produce more heat when under prolonged continuous load, which is generally at its highest when going up hill, being fully loaded, and/or towing (and especially the combination of all three). It is almost impossible to generate compable load by 'pusing hard on flat surface'. Hard acelleration is usually followed by time with your foot off the gas and on the brake (no load). The speeds required to load the engine to the degree it would be going up a hill, foot flat in third, with the OS mother-in-law taking up most of the backseat, is bound to illegal everywhere except on certain slabs of concrete in Germany. Also, at high speeds cooling is enhanced by the hi-speed wind flow though the radiator.
The most straight forward and obivious answer is usually the correct one most (read 98%) of the time. Just looking back at the first post to reply to your problem of your car overheating, the advice given was 'it's most likely the radiator'. Well....may be, but generally the first place to look with either overheating or over-cooling is the thermostat, -because it is its job to control and stabilise the temperature. Hey, if you turn the key and the car doesn't start, thinking "I've got a dead engine management system, or my cambelts snapped" is rather less useful than thinking "do I have flat battery and is there gas in tank"
A thermostat will be easier to remove, test, and cheaper (hopefully) to replace than a waterpump which will also require (apparently) a new cambelt. All the threads I've read regarding waterpump failures in the Twinnie indicate they tend to be sudden and catastrophic (please someone comment on this), although the vanes on the impeller falling off or eroding away isn't an impossibilty. But if you do test your thermostat, you can eliminate as a factor at no cost. Just google thermostat testing I guess, it's pretty simple, but you do have to suspend the thermostat off the bottom of the pot because the metal will be a lot hotter than the water and the thermostat shouldn't be in contact with it.