You will find in the French "Guide des financements France vers le Royaume-Uni" published by the the Science & Technology Department of the French Embassy in the UK, a list (not exhaustive) of existing grants or scholarships (“bourses” being the selected generic term):

Mobility scholarships for a semester or a year of study;
scholarships to pursue studies for example in the framework of a Master;
PhD grants;
research grants;
grants for short stays.

They are classified according to the nature of the structure that proposes them:

French organizations to "go out of France": public sector, academia, associations and foundations;
British organizations which welcome foreigners: research councils, charities, learned societies;
in the framework of specific bilateral agreements;
European Commission: Horizon 2020;
international bodies to go out of France.

For each funding scheme, the following information is specified:

Scholarship type;
covered scientific area;
objectives;
eligibility;
stay duration;
funding level when available;
deadlines;
useful web links and contacts for further information.

Mobility Funding Guide to the United Kingdom