how long after 1066 was it before most "English" landed gentry could truthfully claim to have grown up speaking English rather than French?

I can contribute some information from a History of the English Language class but I won't be able to cite it as it came from my professor (Can provide their name and the university if requested).

Basically French was set as the dominant language among the nobles until around 1204 when John Lackland lost Normandy to Phillip II of France. This basically forced most of the nobles to choose between staying in France or England, the most wealthy were largely based in France as that was where their main holdings were while the lesser nobles stayed behind as most of their land was gained from the campaign in 1066. This basically resulted in the ties being severed between England and France so the nobles that stayed in England no longer had a reason to hold onto French as much as they had before and so began a long process of English taking over the nobility. This is backed up by the fact there was an influx of French word into English at this time as the nobles started picking up the language of their subjects (far more convenient to just learn the language of the locals than keep using interpreters, especially considering French was only spoken by 5-10% of the population).

This basically meant that there was a long period of bilingualism where the nobles could speak non-Central French (Important as at the time Central French was gaining prestige and all other dialects were basically a joke) and English. Then nobles started arising that could only speak English (No need for a form of French that even the French had abandoned) and then eventually French was gone.

Sorry, don't know exactly when English became the primary language exactly but hopefully this provided some more detail about what you were looking for.