The melting of the Greenland Ice cap could indeed lead to the Gulf Stream ceasing, and paradoxically for the British climate to get colder.
The Gulf Stream is driven by thermohaline (heat and salinity) flow and sinks off the coast of N Europe to form the Atlantic Conveyor deep water current which then heads south, and eventually to the Indian Ocean and thence back as surface warm currents via the Carribean and back to Europe as the Gulf Stream.
However, fresh water is less dense than saline water. If the Greenland ice cap melts, it will dilute the salinity of the Gulf Stream, and could reach the critical point where the cooling Gulf Stream water is no longer saline enough (and hence dense enough) to sink and form the Atlantic Conveyor. This will then break the cycle, and the Gulf Stream will cease (for a period).
This is believed to have happened in the past, when the melting of the Laurentide Ice Sheet over N. America resulted in a massive ice-dammed lake (Lake Agassiz). Further melting then released the dam, and a massive amount of water poured into the Atlantic, and for a period the Gulf Stream stopped. This led to a renewed ice age for a time as the flow of heat from the equator to the north stopped (Younger Dryas stadial).