Europe gets it warm climate (relative to how northerly much of it is) because of ocean currents transporting heat from the south. Climate change might cause these currents to change and one of the ways it could change is a shutdown of the currents that bring the heat. This would cause the climate of Europe to become more like northern Canada with a lot of ice, ultra cold winters and moderate summers.
HOWEVER, it is far from certain this would or even could happen. Ocean currents are a pretty complex and chaotic (in a math sense of the word) system. Predicting what effect climate change has is very very hard. It will probably have an effect for sure, but what that effect is would be hard to say. Even a shutdown of the amoc could mean a new current system replaces it and who knows how that system functions.
Especially for a place like London which isn't on the ocean but on the North Sea, it's hard to predict what would happen. If the ocean currents cause the climate around the UK and Europe to become colder, London could still get a lot of warm water coming in from the south in the North Sea and still have a moderate climate.
We should be very scared of climate change. If you think about how much energy you need to put into the ocean to even heat it up a little bit, it's too much to fathom. But we did heat it up, by a lot. However these kinds of speculations aren't really helpful. At best it scares people who are scared to begin with, at worst it has climate deniers tell people akshually the world will become colder not warmer.