I am entering a three-lane road, with the intention to make a U-turn about 300m down the road. Should I wait for all three lanes to be clear before merging?
No. If you wait for all three lanes to be clear, you will wait for a long time and will be holding up other cars behind you.
The right way would be to merge when the lane nearest you is clear and gradually filter to the turning lane.
Even if you miss the U-turn, there will be other opportunities to turn soon after.
Be considerate and do not hold up others.