The term 'global warming' has been more or less retired because of exactly what bb observess - the fact that the average temperature of the earth is rising doesn't mean that we all experience warmer temperatures.
The preferred term now is 'climate change' but that's not great either because it doesn't really communicate how terrible things are likely to get if the current predictions are correct.
We're more likely to see increasingly frequent 'extreme' weather events than just 'it's getting warmer'. It's very hard to predict when and where will get what sort of extreme weather but more destructive winds, more floods and more wildfires are already being observed globally.
Rossall is also correct about the Atlantic conveyer system which is already slowing down and could, theoretically, stop entirely. This would mean the gulf stream that currently protects us and gives us a much milder climate than our latitude would normally get (we're at the same level as Canada and Russia remember), could disappear and yes, that would mean an ice age for the UK and a lot of Europe.
The real problems, in our lifetime, are likely to come from not from the weather events themselves, but from massive numbers of people moving because the area they used to live in can no longer support them, due to water contamination and crop failures.
Large numbers of people trying to move is going to lead to wars, especially if fresh water and food are at a premium.
I'm personally pretty pessimistic about the situation. It's very hard for people who aren't being directly affected by climate change right now to understand the necessity of behaviour change right now. Humans just aren't wired for making massive changes that won't benefit them for decades to come (and if they make the changes, they'll never notice the benefit). We can come up with endless justifications for not changing anything.
We also have systems of governance that are profoundly incapable of making the changes on the scale and at the speed required - any democratic government acting with the appropriate level of urgency will be pretty swiftly kicked out of office, meanwhile any autocratic or dictatorial regimes aren't generally known for doing anything other than enriching themselves and their friends.
Many climate scientist say that its probably too late to mitigate against a lot of the impacts we're likely to see. That doesn't mean we shouldn't try.
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The world is a big place where things happen almost every day.