We use state-of-the-art global climate models and observations to show that the projected higher pressures over the British Isles due to global warming are part of an atmospheric response to the decelerating Atlantic meridional overturning circulation (AMOC) causing a reduction in the associated northward heat transport, keeping the North Atlantic relatively cool. However, considerable inter-model differences in the projected weakening of the AMOC lead to a large spread in the projected wind changes. Hence, the uncertainty in the projected reduction of oceanic heat transport is a main source of uncertainty in projections of Western European climate change. Better-constrained projections of European summer climate thus rely heavily on a more realistic representation of ocean processes in climate models.