The problem is that IIRC from previous threads the Faroe Islanders were, linguistically at least, closer to Denmark than the Greenlanders were, also being generally more conservative and religiously observant as well. Even without those factors I just can't see a few squaddies handing out some bars of Cadburys during the war as being swaying enough as to suddenly make people want to join the UK.
Realistically if you want the Faroe Islands to become a part of the UK then I'd say that you need to look to it occurring before 1900, the tidying up after the Napoleonic Wars being the latest. Whether they'd join the UK proper or become a Crown dependency like the Channel Islands is an interesting idea, if the latter then I do have to wonder if it might not have given the Shetland Islanders ideas in the future.