You sure do ask a lot of questions.

That's ok though, I like answering them.
The quest isn't bugged, don't worry. Just because you accepted the "A Spot of Tea" quest doesn't mean that a carriage will spawn, you might have to wait a few days, or a week to be exact.
The carriages are random spawns, the places I've seen them spawn were:
- The beach by the port east of the Guild.
- Right beside MouseP kingdom on the south side of the map.
- Very close by BirdP kingdom (although this might have been for her medicine quest).
I've never seen the quest have three duplicates though. Interesting.