Stick with it. Despite what BA has told you, this is within their control and I do not see how this could constitute extraordinary circumstances.
At their home base, they could have taken the decision to put you onto another aircraft sooner.
Clearly aircraft should not fly if there are problems but this does not excuse BA from getting passengers to their destinations at the earliest possible opportunity. Using BA's logic, they would deny every compensation claim in the event of a technical issue, which is a nonsense.
And I agree. You could also add the final line of the Sturgeon ruling:
Originally Posted by ECJ
Article 5(3) of Regulation No 261/2004 must be interpreted as meaning that a technical problem in an aircraft which leads to the cancellation or delay of a flight is not covered by the concept of ‘extraordinary circumstances’ within the meaning of that provision, unless that problem stems from events which, by their nature or origin, are not inherent in the normal exercise of the activity of the air carrier concerned and are beyond its actual control.
Nose gear door issues are inherent, normal technical problems which are not extraordinary circumstances. There was more that BA could have done under "the actual control of the air carrier" including routing with alternative services.
Ask them to review the case in the light of these points, culminating in the phrase, "please respond in the next 10 days since I am unable to let matters rest there". 11 days later give them 7 days notice before action.