Civil Partnership Act 2004
The Civil Partnership Act 2004 is an extremely large Act and can be found at http://www.opsi.gov.uk/acts/acts2004/ukpga_20040033_en_1
The Same Sex Marriage Act 2013 can be found here http://www.legislation.gov.uk/ukpga/2013/30/schedule/5/enacted
More details on converting a civil partnership into marriage can be found here https://www.gov.uk/government/news/conversion-of-civil-partnerships-into-marriage