Anything is possible (including using a false name), but you have to hope that wasn't the case , and probably wouldn't be.
Also - don't forget that young mothers could sometimes go to a mother and baby home some distance away to have a child, so don't limit your search to the home district.
Note that if you do identify the child, and the entry is marked as "adopted", you won't have access to anything to give you their adopted name.
In the paper indexes (as used on FreeBMD) an illegitimate birth would normally have the indexed surname the same as the mother's maiden name column (doesn't apply to the new GRO index).