Andy J2022 is essentially correct - (in E/W) when the court issues an adoption order a message is sent to the superintendent registrar who holds the original birth entry for the child and they will add the word "adopted" in the margin and initial it. There is no other indication made on that entry to connect to the adopted identity of the child.
A record is also made in the Adopted Children Register (ACR) held at GRO which will have the details of the child (under their new name, if it has been changed), details of the adoptive parents, and the court issuing the order. The place of birth may just say "England" or "Wales" rather than a specific location to make tracing the original birth more difficult, although later entries do show the registration district. There is no mention on the ACR record of the original birth name. The link between the two documents can only be accessed by an authorised adoption intermediary service.
Certificates from ACR entries can be obtained from GRO, but the index is not available on-line - it is viewable on microfiche at the few designated libraries that hold full copies of the GRO indexes.
As already said, the centralised court run system started from 1927 (in E/W), but that doesn't mean that adoptions before that left no trace - there are often records to find in charity papers, church records, poor law unions etc. although they can be difficult to track down. I even have a copy of an adoption contract drawn up by a solicitor in about 1911 recording the handing over of a child, and listing the conditions agreed between the birth mother and adopted parents about how the child should be brought up.
I'm looking at the birth cert for my wife's aunt born late in 1921 which has an "adopted" note added by a registrar who was in post up to 1930 - so could the adoption be as late as that?
The adopted annotation does mean that there was a court process sometime from1927 onwards, and therefore there should be records somewhere. It was quite common for children adopted before the act to be adopted again through the courts later to provide legal confirmation of the relationship and prevent any issues arising later about inheritance for example. These adoptions could be done anytime until the child was 21, so can be many years later.