When using the new UA Eldritch Invocation, Chain Master's Fury (from the UA Class Feature Variants, p. 10):
Chain Master's Fury
Prerequisite: 9th level, Pact of the Chain feature
As a bonus action, you can command your familiar to make one attack.
When does a Chain Familiar’s attack occur?
The Pact of the Chain Pact Boon (PHB, page 107) states,
Additionally, when you take the Attack action, you can forgo one of your own attacks to allow your familiar to use its reaction to make one attack of its own.
This means that the familiar will immediately use its reaction and make an attack, replacing one of the attacks during the Warlock's attack action. However, the Chain Master's Fury invocation makes no such statement that it requires any action on the familiar's part. In other words, it is a free action for the familiar.
Immediately, as part of your bonus action, the familiar makes an additional attack. This leaves your warlock open to use his action for something else like Eldritch Blast, or to make two attacks if you have the Thirsting Blade Pact of the Blade Warlock Invocation or Extra Attack feature from another class.
The text for this Boon says simply:
As a bonus action, you can command your familiar to make one attack.
There is no mention of a delay, nor an additonal cost to the familiar, therefore there isn't one.
For comparison see the Ranger Revised's beast companion (also Unearthed Arcana):
The 5th level ability Coordinated Attack states:
When you use the Attack action on your turn, if your companion can see you, it can use its reaction to make a melee attack.
In contrast to the Warlock's familiar, the beast companion has to use its reaction to make the attack, and so that is clearly spelt out.
As an aside, if you're worried about balance, if anything, the Ranger's is the more powerful action - it's likely that your PC's bonus-action (the cost the warlock pays) will be more valuable to you than your beast companion's reaction (the cost the Ranger pays). However the Ranger does also have to pay the opportunity cost of needing to take the attack action themselves - whereas the Warlock doesn't need to do anything to trigger this use of their bonus action.
That said, both of these optons are unearthed arcana, and thus playtest material, and so may well be unbalanced when compared to official published game materials.
