Warlocks in the Chain Pact have access to an invocation called Chain Master:
Voice of the master of chains
Prerequisites: Chain Pact
You can telepathically communicate with your pet and perceive through his senses as long as you are on the same level of existence. In addition, while perceiving through the senses of your pet, you can also speak with your pet in your own voice, even if your pet is normally unable to speak.
One of the limitations of the Wild Shape feature of the Druid is:
You can not cast spells and your ability to speak or take any action requiring hands is limited to the abilities of your dumb form.
If a Druid / Warlock multiclass had this summon, could they speak through their familiar even using Wild Shape to adopt a form that has no language?