How to enable British English spell checker from French version?

From what I understand, the spellchecker dictionaries available depends on what language you currently set the portion of text. You can only set one language for a specific portion of the document, so you'll have to set what is French and what is British English. This is what triggers the spelling dictionaries used on it. To do so, select the text or paragraphs, then go to tools > languages and set language, using either for selection or paragraph.

If desired language does not show up, select "more" and you'll be brought to a dialog for specifying the language based on whether the text is western (which where both languages mentioned are), Asian, or complex text layouts scripts. It should be noted that languages listed here does not necessarily mean you have a dictionary for it. It is used to specify the language of select content, which triggers what dictionary, if one is available for language, is used for spellchecking. When selecting the language, if any have a spellcheck icon, you have the dictionary for said language, otherwise you do not and will not spellcheck. I'd strongly recommend seeing if their is spellcheck icon next to English (UK) in the drop down from this window to be certain you have the dictionary.

I don't know whether you installed the the first or all "languages version" as the first installer excludes some dictionaries and templates for certain languages. I'd strongly recommend seeing if their is spellcheck icon next to English (UK) in Tools > Languages > whatever window in the language select drop-down for Western to be certain you have the dictionary. You'll have to trigger the drop down listing and scroll to desired language to be able to see the spellcheck icon. If set language does not have dictionary, it won't attempt spellchecking when set to this language.

Note: Sometimes LibreOffice can auto detect words that may not be in a specified language and may suggest another language in such cases, but rarely do I see this get triggered. It is often up to the user to manually specified the language of the section or selected text to ensure any dictionaries are properly used for spellcheck.