If I have touched the gold part of the CPU (Intel LGA775), how do I clean the processor?

You don't. It can be too easily damaged. And hopefully you did not destroy the CPU already by mishandling it. Never touch the contacts of any computer component - CPU, RAM, add-in card, etc. Simply by bringing your fingers near the contacts can cause an electro-static discharge (ESD) (static shock) from your fingertips to the device. The discharge can be so small that humans cannot even detect it, but still of such potential to easily destroy ESD sensitive devices - in particular CPUs and RAM modules.

So this is not good.

They use gold because it does not oxidize (like rust or tarnish). Oxidation creates a barrier (resistance) to the flow of electrons - not good in high-speed digital circuits. Skin oils on electrical contacts promotes corrosion and attracts dust, creating resistance.

I recommend leaving it alone and hope all is okay, or taking it to a shop and let them clean it up. They have the necessary ESD control and cleaning solutions.

Thank god I didn't destroy it. I guess is because UK is damp and we don't get as much static shocks here.

While it is true that ESD is less of a problem in a humid environment, it is important to note that cool air is a by-product of air conditioning - with "conditioning" being the removal of moisture in the air. And since ESD sensitive devices are way more sensitive than human awareness, it is still a serious issue even if you live in a high humidity climate area, it is the operating environment (ambient conditions of the room) the computer runs in that matters.

If your CPU was not damaged after touching the CPU's contacts, then you are correct - you have a Higher authority to thank.