There was no legal adoption until 1927 (Family Records Centre index starts from this date) - the act was passed in 1926. From that date those children legally adopted had the same rights as a 'sibling' who was the natural child of married parents (children born out of wedlock even if their parents later married still remained (in legal terms) illegitimate until a similar act also of 1926.
Families brought up children as their own before the act or children were 'fostered with couples. In the past or today - you can change your name to any name you want. Today all you need to do is inform all the necessary authorities that in future you wish to be called for instance 'Screaming Lord Sutch'. You don't need to change your name by deed poll - actors do it all the time and so do women in particular, usually after a divorce when they might choose to revert to their former maiden name, or if they are living with someone but not married to them, but wish to be known by their name - less of an issue today than it was in the past.
Regards
Valda