Since I added a string of perhaps opinionated but very fact based comments on other people's answers, I thought I should add my own.
3. Both are in use.
In my comment on the Q I said this would be unlikely to happen for any given part of the population at large (as in Canada) but in terms of governance, it is the most sensible approach (as in Canada), because it means the population at large could, in all places, officially communicate with, and understand, the local ruling elite.
To be clear, I don't mean that the governing "political/academic/military" class would as a rule be bilingual (and it is a great stretch of the imagination to claim that in Canada). I mean the members of such a class in an Anglo-French Empire would not adopt one language exclusively for governing the entire Empire. So, contra the currently top voted answer here, I do not think the aspirations of a far away central court could prevail for long if that meant enforcing official use of only French over all colonies regardless of what language was spoken predominantly by the immigrant population. By "immigrants" I mean French or English natives who have populated outlying parts of the Empire.
Part of my reasoning is that for such colonies to be successful, unless immigration to them is forced (e.g., as with prisoners sent to Australia), a certain amount of respect from and integration with the ruling elite must be in evidence or you would not have many colonists participating in an "Anglo-French Empire". So this is distinct from areas where the Empire is really military outposts presiding over a mostly native, non-French, non-English population, where no pretense of respect or integration would be necessary -- those people have no where to go, and if you can dominate them by force and conduct official business in a foreign language native to the ruling elite, there won't be any issues the "force" part doesn't cover.
Because of this logic, I don't think the ruling elite of the Empire would for long aspire to using only one language or the other; they might in fact come into conflict with one another on the issue but presuming the more competent prevail (without which the Empire would be short lived), they would quickly realize that an official mix of French and/or English, appropriate to the local population, would be most expedient.
This would require that the imperial government everywhere would need at least a few people capable of translating from one language to the other, and that communication between regions/persons could be conducted in either or both languages.
Native English Lord A could send native French Lord B messages in English, confident that B or one of his retinue could deal with this.
French Lord B could reply in English or French.
Native English Lord C, who is proud of his French, might choose to use French with everybody all the time.
It's worth noting that in Canada, getting many/most(?) federal jobs requires proof of bilingualism. However, it's also worth noting that one thing which qualifies as proof (for a native English speaker) is having completed French to the end of high school -- which sounds like a long time, but it does not, in fact, mean that everyone who did so can actually speak or understand French, particularly once they've been out of school a few years. So the requirement, and the qualification, are often a pretense. I mention this as an example of how aspirations of this sort don't prevail, reality does, and to make an empire work over the long term you do need to take the latter into account.