Oddly enough, of all the aspects and abilities of your favorite pro players, one of the easiest to attain is speed.
Mostly, it requires lots and lots of mindless repetition. If you are not afraid to play the same riffs and phrases and scales repeatedly, you can play fast. A few weeks of correct practice can make a huge difference in your speed.
Except in the case of the Warble, always move the harmonica and not your head when you play. Check in the mirror to be sure you are doing this correctly.
Relax. This is by far the most important aspect of increasing your speed. Again, check in the mirror to see what is tense when you play. There should be no tension any where in your upper body including your shoulders, neck, and face. Within the face area, keep the tension away from from your eyes, forehead, mouth, and chin. It is OK when working on your speed to drop your volume a bit to maintain relaxation. Once you've increased you speed, you can go back and try to play louder and stay relaxed while playing faster.
DRILLS TO DO:
Blow-Draw pattern on hole 4 (diatonic harmonica).
Warbles on holes 4 and 5 draw or 3 and 4 draw.
Scales: Major, Blues, Country.
Triplets: Any that you know.
Tonguing: Repeating the same note faster and faster. This is not Tongue Blocking.
Riffs: Any favorite old, completely memorized riffs.