If a human druid (30 ft/round), in combat/initiative, uses all their movement speed, than wild shapes into a brown bear (40 ft/round), do they get to move again? If not, what is the formula for their remaining speed if they had only used half of all their movement?
Treat this as a case of multiple speeds
The Player's Handbook (p190) states:
If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed.... If the result is 0 or less, you can't use the new speed during the current move.
At any time during her turn, the druid's total distance covered can be no more than her current movement speed. Wild shaping does raise her movement speed, but does not affect her accumulated movement.
In your hypothetical example, the druid moves 30ft as a human. Then she wild shapes into a bear, and now has a movement speed of 40ft. She can therefore move a further 10ft after shaping.