Hi baobob,
I've travelled in First with one infant and one small child, so not quite the same.
The main issue you will face is the New First and Old First. The Bassinet/Britax positions are different for the new and old cabins. For the old cabin on the 777 they are definitely 1A & 1K - problem for you is that they are on opposite sides of the cabin, so you won't be sitting close to each other. In New First it is apparently 4F (though have never seen it set up) on some sort of shelf and is the only cot position available, so not sure how you are meant to do two there.
For 747 - 1A & 1K Old First and are close to each other. Don't know on the 747 for New First.
Re: Strollers - we managed to keep ours with us and the crew put them in the cupboard. Not sure if a double stroller will fit though. Bear in mind though that if you are in transit at LHR and the stroller was checked you won't see it in LHR, so need to make very sure that you have it there.
Also, make sure you book a Cabana ahead of time in the Concorde Room at LHR before your flight as this will give you a nice private room to freshen up, relax, and change the kids before the flight.
As for oxygen masks - sorry, no idea at all.
Good luck with the trip ^