Regarding the rail trip from Lucerne to Interlaken Ost, the shortest time I could find was 1H:50M, which seems to be consistent for every trip on that route. There appear to be departures hourly. Add about 25 minutes for the trip from Interlaken to Lauterbrunnen (you'll have to change to the smaller Berner Oberland train at Interlaken Ost). More @Wikipedia
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