There is no obligation for a woman to change her surname after marriage in the UK. Getting married doesn't change her name. Getting married gives her the option to change her surname.
If a woman chooses to change her name, it is probably more practical for her to change it on all official documentation, but it is not obligatory to do so.
For example, if she chooses to change her surname, she can send her current passport (in her maiden name) to the passport office along with her marriage certificate and an application form. She will then receive a new passport showing her married name.
If she chooses to change her surname on her driving licence, she does the same thing - she sends her current driving licence, an application form and [a copy of] her marriage certificate to DVLA (the licensing authority) and they will issue a new licence with her married name.
Most official changes must be made by the individual woman by writing to the relevant authority (bank, tax office etc) and enclosing a copy of her marriage certificate.
These days, quite a lot of people choose to change their name for the private life, but not for their work life. If you have worked for a company for several years, and everyone knows you as Julie Smith, it's practical to keep that name at work. There is no obligation to tell your employer that you are now married and you do not have to change your name at work.