In order to register a marriage in Sweden or change your name, you need to contact the Swedish Tax Agency (Skatteverket).
Swedish citizens living in the UK must apply for a "Certificate of no impediment" (hindersprövning) at the Swedish Tax Agency, more information here.
The certificate "No trace of marriage" is needed if it has been more than 12 months since you registered as living abroad. The certificate is issued by respective General Register Office in England and Wales and in Northern Ireland and by the National Records i Skottland.
British citizens living in the UK must submit a "Certificate of no impediment" in order to get married in Sweden, more information about this here.
Swedish citizens living in the UK have to give notice to their local Register Office, more information here.
Swedish citizens that have been residents in Sweden can order an extract from the population register (personbevis) that shows their civil status at the time of leaving Sweden, more information here.
The Swedish Church in London conducts wedding ceremonies according to British law, more information here (in Swedish). The Embassy does not conduct wedding ceremonies.
Swedish citizens living in the UK who are getting married in another country, including Scotland, can apply for a marriage license at the Embassy of Sweden in London, read more here.