I usually use chocolate letters on my cakes but wanted to try using royal icing. How long would it take for the letters to dry if I piped them out onto wax paper and put them on the cake? Would they dry overnight?

Yes they should be fine as long as you put them somewhere warm and dry to set. For extra gloss put them underneath a reading lamp for 10 mins then I would allow 12 hours

If you are really desperate for them quickly you can put them in a really low oven with the door OPEN. I have managed to get usable letters in a hour or so with this method but it is tricky and you have to be very careful that the letters don't start turning colour in the oven

I like the reading lamp trick MissBaritone. Really nice tip. Thanks