Thanks for your question.
There is no UK body which would hold a copy of your birth certificate. In all probability you will have to contact the local or regional registry in Uganda, or the closest one to where you were born. I understand that your parents would have had to register your birth with your local council chairman for the area where you were born, so you should speak to your parents or local council back in Uganda
The local districts of Uganda are laid out on this page, select the one you want and then contact the local government branch. They should be able to refer you to the appropriate department.
Alternatively, the Ugandan High Commision in London should be able to tell you where the registry in Uganda is but they will then ask that you contact them directly. The contact details for the Ugandan High Commission are on this page:-
The actual website for the Ugandan High Commission is here:-
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