Because the last ability modifies the rules of the game, it affects not only planeswalkers you control when it resolves, but also planeswalkers that come under your control later in the turn.

For the first ability, it doesn’t matter whether the planeswalker is still on the battlefield as your end step begins. If you activated one of its loyalty abilities that turn, The Chain Veil’s triggered ability won’t trigger.

Each additional time The Chain Veil’s last ability resolves will allow you to activate a loyalty ability of each planeswalker you control an additional time. For example, if you activate The Chain Veil’s last ability, untap it, then activate it again, you can activate a loyalty ability of a planeswalker you control three times that turn.

After the last ability resolves, you’ll essentially be able to activate a loyalty ability of each planeswalker you control a total of twice during your turn. The timing rules for when you can activate loyalty abilities apply each time; it must be your main phase and the stack must be empty.

The second loyalty ability you activate doesn’t have to be the same as the first ability. For example, you could activate a planeswalker’s first ability twice, or you could activate a planeswalker’s first ability, then activate its second ability.