Final answer:
Barbados is known as The Mother Colony of the West Indies, significant for its early sugar production and use of slave labor, setting a model for other British colonies.
Step-by-step explanation:
The country that was the first home to British and French colonies in the Caribbean and is thus called The Mother Colony of the West Indies is Barbados. By the mid-1600s, Barbados had become one of the most essential English colonies due to its sugar production. It was the first English colony where enslaved people were used extensively, setting a model for other English slave societies in the Americas. Barbados's sugar industry was particularly significant because, during the 17th century, Caribbean islands like Barbados and Jamaica were among the most profitable British colonies due to their output of cash crops such as tobacco and sugar.