It’s not just because of the gulf stream, and the gulf stream isn’t going anywhere anyway. For one, Europe is a heavily maritime continent with few massive land areas on the scale of North America or Asia, which contributes greatly to the temperate climate, moreso than the effect of the gulf stream. If you’re surrounded by water then it takes a lot more energy to heat the area up significantly or cool it down. That’s why Siberia and Canada both have more extreme temperatures than most of Europe, hot and cold.

Furthermore we don’t actually expect the gulf stream to collapse totally - some of the larger circulation that the gulf stream is a part of will certainly destabilise and collapse (the northern stretches of it beyond the coast of Ireland particularly), but the gulf stream itself is driven by prevailing winds coming off North America which are driven by the relief of that continent (e.g. the Rockies). So we don’t expect the gulf stream to go away unless the Rockies go away too, and this means that the heat transfer from the tropics to the North Atlantic that it generates won’t go away either. The heat transfer might become a little less efficient over time but that effect should be counteracted by, well, global warming itself. We will probably see more extreme winters and summers in western Europe over the next century (I mean, we already are) but it’s not going to be the Day After Tomorrow style event that the media likes to imagine.