I suppose it depends what you mean by fucked. Will the human race be wiped out ? I doubt it.
Will there end up being a lot less of us ? Probably.
I'm pretty sure the UK is well placed to handle climate change. We are overpopulated, so scrub off half the population, but I doubt whether in climate terms it will get too bad. it might even be quite good to have a warmer climate here. We can grow wine and other crops. The sea level will rise but we can either accept the decrease in land available or build flood defences to claim it back.
Unfortunately we don't live in isolation, so unless we make ourselves pretty much self sufficient, we'll be affected by the wars and chaos that happen everywhere else in the world as a result of it.
The modern world is supported by massively complex supply chains, so what will happen in climate change is that countries will turn increasingly inwards to focus on their own problems if the situation there is not too bad, or invade others if it is.
Since a lot of the products we rely on these days are made of hundreds of thousands of components from hundreds of different countries its highly likely the future won't be about gadgets or anything complex, but back to the simple things, that is if the nuclear war doesn't wipe us out as people get increasingly desperate.
Human beings are just too short term in their outlook to deal with these sorts of issues. Ditto many of the issues associated with nuclear power. So the issue will deal with us. The earth has been through all sorts of nasty stuff in the past. Massive periods of volcanism, snowball earth. Multiple mass extinctions. Asteroid impacts. Each time life bounces back in a few hundred million years.
No one is going to stop saying they have a right to have as many kids as they want. No one is going to stop going on nice foreign holidays. No one is going to stop eating meat. No one is going to turn the heating down. You only have to look at the threads on here to understand that. People think they are entitled to these things, and that putting a yogurt pot in the recycling somehow cancels out all of the environmental impact of their lives.
We're rapidly using up all the agar gel in the petri dish and we're going to hell in a handcart, so best to enjoy life while you can. It's not something I can stop. So I don't worry about it. I'll be dead by the time it all kicks off.