I may not be able to answer very well in regards to the British, but as for the American that might be another issue. Have you tried to go to the nearest military base near you? Someone in the headquarters may be able to point you in the direction you want in order to obtain the records for that country.
Since the States and Britain are friendly to each other, the base near you again might be able to give a clue. If not, check into the British website and ask there for the email or mailing address if someone here can not provide an exact address for both. I have heard of a place in Washington D.C. that you could get in touch with for proof or denial of military service for the American services.