In American English, the "l" is produced in "fault", as is the "r" in "world". I can understand that these may be less pronounced in British English.
In the word "doubt", the "b" is not pronounced because it comes after another labial-type (lip) sound. In this word, the preceding sound is "oo" (similar to a sustained "w" sound), in which the lips are protruded. Because "b" is also a labial (lip) sound, it is omitted. Yet, the "b" remains present because of the Latin root from which the word stems: "dubius" (from which we get the word "dubious").
This, by the way, is also why we see a silent "b" in other words:
lamb, limb, climb, numb, thumb, tomb, etc.
The preceding "m" is a labial (lip) sound, so the "b" (also a labial) is omitted in pronunciation.