Hello Stu,
To the NW Atlantic, it looks indeed like the cold East Greenland current follows the coasts: see http://en.wikipedia.org/wiki/East_Greenland_Current
To see the direction of the streams, I put the position of Titanic sinking on two map extracts.
No too readable, but anyhow, one can infer that indeed the iceberg must have taken the path of the Gulfstream, after having come from the North.
The first extract is from Wikipedia, the second of the December pilotchart 106.
Regards,
Luc