Let's break this down in to Rules As Written, then I'll give a nod to Rules As Fun. But the important thing to start with is this: D&D is not a Physics Simulator (alternately, it is a hilariously bad physics simulator).
Rules As Written
The rules for falling are as follows:
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.
That's it. No modification based off size exists. Yes, realistically, a small enough animal would have a low enough terminal velocity to avoid harm--but D&D has no consideration for that.
As for the time it takes to fall, the rules are somewhat expanded upon in Xanathar's Guide to Everything (p. 77):
The rule for falling assumes that a creature immediately drops the entire distance when it falls. But what if a creature is at a high altitude when it falls, perhaps on the back of a griffon or on board an airship? Realistically, a fall from such a height can take more than a few seconds, extending past the end of the turn when the fall occurred. If you’d like high-altitude falls to be properly time-consuming, use the following optional rule.
When you fall from a great height, you instantly descend up to 500 feet. If you’re still falling on your next turn, you descend up to 500 feet at the end of that turn. This process continues until the fall ends, either because you hit the ground or the fall is otherwise halted.
Here, these optional rules state that the moment you begin falling, you immediately descend 500 feet. No consideration is given to acceleration, you just immediately drop 500 feet, then do it again at the end of your next turn.
Thus, per the Rules as they are written, it does not take time to fall, so (unless you have something like Feather Fall that explicitly interrupts a fall) there is not time to do anything at all before you hit the ground, unless you are falling more than 500 feet and your DM is using the optional falling rule.
And even if there was time...
Wild Shape can only be used on your turn
The rules for Wild Shape are as follows:
you can use your action to magically assume the shape of a beast that you have seen before.
And actions are taken on your turn:
On your turn, you can move a distance up to your speed and take one action.
Thus, unless it is the druid's turn when the aarakocra drops them, they can't Wild Shape... because Wild Shape requires an action, and you can only do those on your turn.
How to Wild Shape when dropped anyway.
An aarakocra that has grappled you cannot cross 125 feet in a single round:
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
So, assuming the aarakocra (which has a fly speed of 50 feet) uses its Action to Grapple you, in a single turn it can only lift you 25 feet off the ground. It may then Dash on subsequent turns to cover 50 feet per turn. Thus, it would take 3 full turns for the aarakocra to lift you to your intended lethal-drop height.
During this time, the druid may opt to use the Ready action:
First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger [...] When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
So, on their turn, the druid declares that they wish to Ready an action to Wild Shape into a mouse if the aarakocra lets go of them. So, the aarakocra lets go, and the druid activates their reaction to Wild Shape (which happens immediately after they are dropped (the trigger), thus interrupting the fall).
But, RAW, it won't help...
The falling rules have no allowance for variable terminal velocity. A falling dragon and a falling mouse take the same exact fall damage.
Rules As Fun
Situations like this, especially as you've described it, are one of the reasons why DMs are fully permitted to ignore, alter, or make up rules on the fly. Per the description of the role of the Dungeon Master in the DMG (p. 4):
The D&D rules help you and the other players have a good time, but the rules aren’t in charge. You’re the DM, and you are in charge of the game.
In a case like this, I would want to reward the player's creativity and roll with it. Ignore all the nitty-gritty rules that are (for the moment) interfering with fun, and let the player have a creative solution to this problem.