Can I use the Ready action on my turn to specify that I will make a melee attack once in range of an enemy, use my mount to get to the enemy, attack the enemy with one melee weapon, and then also attack with my second weapon even though it isn't my turn anymore?
No, because you do not have a bonus action.
The section on two-weapon fighting states (PHB 195, emphasis mine):
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.
However Ready only allows you to ready an action, but not a bonus action (PHB 193, emphasis mine):
First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.
So unless you have some other feature that allows you to take a bonus action outside of your turn (none of which are known to me), the default rule for bonus actions applies, which states (PHB 189, emphasis mine):
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action.
Two-weapon fighting is such an ability that allows you to take a bonus action, but as per bonus action rule, this only works on your turn.
This means to me that if you choose to ready the attack action, you are taking the attack action in response to that trigger. But you cannot take a bonus action to attack with your second weapon, because it is not your turn.
It depends on whether you control your mount or not.
When you get on a mount, you have two options: control the mount, or allow it to act independently. Most of the time, it's assumed that you are controlling your mount, especially when you're riding something like a domesticated horse. Whether you can attack with two weapons on a mount depends on which you do.
If you let the mount control itself, you can't. When the mount controls itself, it has its own position on the initiative order, and does whatever it wants. Assuming it charges towards a foe like you want it to, you can ready a single attack. When you ready an action, you can only take one action when your ready trigger happens. Making a normal attack is one action, but also attacking with your off-hand takes your bonus action. Since you don't get a bonus action while readying, you can't attack with two weapons if you let the mount control itself.
If you control your mount, you can. When you control your mount, it acts on your initiative count, and does exactly what you tell it to do. You can command it to charge on your turn, and make a regular and off-hand attack using your action and bonus action just like you could if you were on foot. You don't need to ready an action at all, you can just act normally.
You can find the rules for Mounted Combat that I reference above on Page 198 of the Player's Handbook.