Legally, adoption wipes out all rights and responsibilities of the birth family,
and substitutes a completely new relationship with the adoptive family. So a
child adopted by a married couple (the only type of couple that can adopt)
becomes a child of their marriage.
Hence a gift by Will to "my (unnamed) grandchildren" will include an adopted
grandchild but not a grandchild who has since been adopted. It would be
different if the child was named.
Similarly, adoption will wipe out a maintenance order or child support
assessment, which is why step-parent adoptions have suddenly become popular
again. It would not, however, wipe out existing arrears.
Most importantly, adoption extinguishes all parental rights to contact with
the child. Court Orders, creating parental rights and regulating them, all fall
away. In theory, when making an adoption order a court can impose conditions,
such as requiring the adopters to allow limited contact or on the child's
religious upbringing. In practice, such conditions are rarely imposed
particularly as there is considerable doubt how they might be enforced.
Once the new parents have adopted a child, they gain all the parental rights
and responsibilities, including, of course, the right to live in whichever
country they please.