Make sure you have the right big spanner, and are equipped to drain the tank.
http://www.tlc-direct.co.uk/Main_Index/Water_Heating_Index/Immersion_Heaters_4/index.html
Do not drain the tank until you have made the heater turn a little in the undo-ing direction, you may need to hammer the spanner to crack the seal, and the mass of water stops the cylinder from creasing as you do this.
Then drain the tank, and remove the element and thermostat as an assembly. (This is like those diy books where it says "unscrew the flywheel nuts and place the fly wheel to one side".
What is meant is "spend an afternoon applying progressivly more unreasonable force, bending your best tools and skinning your knuckles, then when it finally comes free, remove and place the flywheel to one side"....)
Take the old one to your plumbers merchant and ask for a 'long life 'or "incalloy" version of the same thing. Stay clear of anything witha short guarantee period.
this sort of thing is OK.
http://www.tlc-direct.co.uk/Main_Index/Water_Heating_Index/Immersion_Heaters_1/index.html
The thermostat itself can be removed 'dry' as it sits in a tube that goes into the water volume, I'm assuming the heating element is open circuit.
For the last 20yrs or so, immersions have been recommeneded not to be on the same circuit as anything else, as they mop up a lot of the capacity of the ring, but if you have not expereinced nuisence tripping yet, its not a big problem, and not the cause of your element failure. However, when the chance arises, yes, split it off onto its own circuit. A 16A MCB will be fine.
Do not overlook the possibility of a loss of sacrificial anode - it makes a huge difference to the rate at which things rot.
Use PTFE tape on the new one, and be careful not to damage the threads in the tank itself.
regards M.