Everyone loved hearing her say stuff like that, even though no one in the US audience would ever say it that way

Yes, slithy. Maybe when Now Voyager came out people in England would actually have said "don't let's", but it sounds quite old fashioned - albeit quaintly upper-crust - these days.

The phrase has a touch of public schoolgirl about it somehow, which has (naturally) led to its occasional abuse by a certain type of older Englishwoman. Cue that wonderful phrase "Mutton dressed as lamb".