In the UK you have the legal right to be known as and addressed by whatever name you like. You can therefore take and assume any legal name you wish to. A legal name being the name by which you are generally known and called by.
This has been instituted in law in a number of ways.
Cowley (Earl) v Cowley (Countess) 
In the case of Cowley (Earl) v Cowley (Countess)  A.C. 450 - in the House of Lords Court of Appeal, a woman’s continued use of the title Countess was heard. Countess Cowley, continued to use the style and title of Countess despite divorcing and losing her legal claim to it. In this case it was held by Lord Lindley that;
"Speaking generally the law of this country allows any person to assume and use any name, provided its use is not calculated to deceive and to inflict pecuniary loss"
Davies v Lowndes (1835)
Similarly in the case of Davies v Lowndes (1835), Chief Justice Tindal held that the defendant Mr Lowndes' change of name to Selby was valid. Furthermore he held, the simple act of adopting and assuming the new name made it so.
"It has been more than once asked by a learned gentleman of the grand assize, whether the name of Mr Lowndes has been changed in the way which the law prescribes. In this will the condition is, that Mr Lowndes changes his name to Selby. It appears, that at first he retained the name of Lowndes, while the receivership was going on; and that afterwards he took the name of Selby in addition to the other; and I am not prepared to say that that was not changing his name: but at all events he afterwards changed it entirely, and left out the name of Lowndes. There is nothing in the will that purports that the condition is to be executed in a very limited or precise time; therefore, though he took it a little later, and though in some particular acts he might use the other name, it would not at all interfere with the general act of changing his name. And there is no necessity for any application for a royal sign manual to change the name. It is a mode which persons often have recourse to, because it gives a greater sanction to it, and makes it more notorious; but a man may, if he pleases, and it is not for any fraudulent purpose, take a name and work his way in the world with his new name as well as he can"
Structure of a Legal Name
Your legal name comprises two key elements, your first name (forename) and your last name (surname). Each name has a legal designation.
For most people, the title does not form part of the legal name, this is the case for social titles such as Mr, Mrs, Miss, Ms and Mx. There are exceptions to this rule in the case of titles of nobility. For more information on which titles do and do not form part of a legal name click here.
For example, John Smith would be the legal name, whilst Mr John Smith would simply be the social form of address. As above, but with an exempted title of nobility, someone who held the title of Countess would hold it as part of their legal name
Whilst it's not a legal requirement to have both a first and last name, the law tends to presume that you do. Since a person is always known by or called something, whether officially or otherwise, they have a de facto name. Since we have established that there is no law requiring a person to have both a first and last name, it stands that a person may have a first name only.
Changing the name of a Child or person under 16 years of age
The law in relation to changing a name by deed poll differs dependent upon the age of the applicant. Deed poll declarations can only be made by someone over the age of 16.
A person with parental responsibility (PR) or legal guardianship (LG) can apply by deed poll to change their child’s name.
All persons with parental responsibility (PR) or legal guardianship (LG) must approve the change of name for it to be realised. This is normally done by way of written letters of approval attached to the Official Deed.
For more information on changing a child's name click here.
The registration of births in the United Kingdom is undertaken differently by the devolved Parliaments of England, Wales, Scotland, and Northern Ireland. However, each country requires that parents register the birth of the child within a prescribed timeframe. Each of the devolved governments also requires parents to declare truthfully the name by which their child will be known.
It should be noted that a birth certificate is not a form of identification. A certificate of birth merely records the name of the child on the day that the declaration is made. Each devolved government allows for names to be taken and changed after registration, notably in the case of baptism names.