If by "wobbly", you mean that it's not firmly clamped onto the CPU, that's likely to be the problem.
I wonder what sort of CPU cooler it is? Your specs list an I7-8700, with a "standard" Intel cooler. But the specs for the I7-8700 was not sold with a stock cooler. Was one from an older CPU re-used?
If memory serves, some Intel CPU coolers just pushed in place. (I've never used one; I always bought a third-party cooler.) I wonder if that retention mechanism would be damaged if the cooler wasn't removed from the motherboard carefully enough?
I'm in the US, so not best able to suggest a UK source for a new cooler. This one is probably more than adequate:
By the way, thanks for providing all that detail in "My Computer".