Go to the website of the relevant examination board (or of the board which replaced the one which set your exams) and look for information about 'replacement certificates' or 'statements of results'. You can find the website from this list:
http://www.qca.org.uk/qca_4521.aspx
Be prepared to be charged a fairly hefty fee for the service. For example, if you took all of your examinations in the same 'session' (e.g. the summer examination period of 1984), AQA make a charge of �34.50. If, however, you took GCSEs in the summer of 1982, a GCSE resit in January 1983, extra GCSEs in the summer of 1983 and A-levels in the summer of 1984, you'd have to pay four lots of �34.50 (=�138.00) to get your results from AQA.
Chris