Your first cousin's grandchild is your first cousin, twice removed.
Your second cousin's grandchild is your second cousin, twice removed.
If you start looking a grandchild of a first cousin once removed, a third cousin twice removed, etc., the answer becomes more complex and depends on whether you are a member of teh younger or older generation in the relationship.
The grandchild of your grandmother's first cousin is your third cousin. The grandchild of your grandmother's second cousin is your fourth cousin. The grandchild of your grandmother's third cousin is your fifth cousin.
Second cousins.
The grandchildren would be the parents, aunts and uncles, and first cousins once removed, of the great grandchildren.
Your grandchild and your niece are first cousins, once removed.
Third cousins once removed.
You and your uncle's child are first cousins to each other. You and your uncle's grandchild are first cousins, once removed, to each other.
Your child and your sister's child share a common grandparent and are first cousins. Your grandchild and your sister's grandchild share a common great grandparent and are second cousins. There is a generation out of synch, so they would be second cousins once removed.
They are cousins.
You and your first cousin's grandson are first cousins, twice removed, to each other.
Your child and your nephew are first cousins, so your grandchild and your nephew are first cousins once removed.
You would be Second Cousins
Second cousins, once removed.