Found this snippet of information on the internet - original dated 2009 so don't know if it still applies today.
When an adoption has been granted by a court in England and Wales, the
court issues an order which is the authority for an entry to be made in
the Adopted Children Register held at the General Register Office.
Adoption certificates are then produced from this register under the
child's new details. The adoption certificate replaces the original birth
certificate for legal and administrative purposes.
The original birth record is not removed following an adoption, however,
the entry in the birth register is annotated with the word "adopted" in
the margin and all subsequent copies of the entry will show the
annotation.
It is possible to obtain copies of adoption certificates from the General
Register Office when sufficient particulars about the adoption are
supplied to enable the entry in the Adopted Children Register to be
correctly identified.
The information contained on an adoption certificate is as follows -
* The name of the adopted person, the date and location of their
birth and the sex of the child
* The name, address and occupation of the adopter(s)
* The date of the adoption order plus description of the court who
made the order
* Signature of the officer at GRO deputed by the Registrar General to
attest the entry
There is nothing on the GRO index to indicate an adoption so I would imagine that any re-registration would appear on the Adopted Children Register held by the GRO which presumably is not accessible to the public.