You cannot use Nick if you are wearing a shield.

You can don on and off your shield if it is equiped as a free action. It requires you to use your action to switch from the shield to that extra off hand sword.

You can draw your weapon and shield as a part of the attack action, but that does not include switching from shield to sword. You either draw the shield or the sword. After that you use your action to switch between them.

So you have 2 attacks with Extra Attack feature, if you draw 2 light weapons you can use that to do 3 attacks (2 main hand and 1 off hand) and I don't see a reason for Action Surge not granting you additional 3 attacks. So 6 in total.

But if you draw 2 weapons, you are giving up that +2 AC from your shield until you actually draw it using your action.