There will not be any dual honours degrees between medicine and French. You clearly don't need to pursue a degree in languages just to learn a foreign language, though.
Most universities will provide cheap/subsidised language courses that are available to students - Oxford (for example) charges students £38/term for weekly classes (2 hours each). They will be available at all stages from beginner to fluent. There's no reason why you shouldn't register for these, which are either in the day or in the evenings. Some medical schools will let you do this in lieu of a Special Study Module. You could then spend your elective in a French-speaking country (Canada, French Guyana, West Africa, etc) if you were keen.
Universities are pretty solid places for learning anything you want - there will be a French Society, possibly undergraduates studying language degrees, and French newspapers/books in the library.