Are there any rules of which name I can choose?

There aren't exactly any laws in the UK which regulates which name you can choose, though it's always best practise to choose something which will be accepted by government bodies. For example, the UK Passport Office set their own guidelines on which names they will accept for a passport application and will not issue a document with a name which is:

• impossible to pronounce
• vulgar, offensive or blasphemous
• promoting criminal activities
• promoting hatred including racial, religious or homophobic
• showing ridicule for any groups, government departments, companies or organisations
• a title which may be deemed misleading such as Baron, Lord, etc.