The balloon wreckage probably floated with the wind down in a different direction then James Bond. Anyways the movie focusing on James Bond not a exploding balloon, so the directors or whoever left out the balloon and zoomed in on James Bond.
I do not think that the British Government would spend billions of pounds on such an important tourist site whose roof/cover is not fire proof. Granted we do not see the wreckage but thats because we mostly get close ups of Bond/Stunt Double falling down the Millennium Dome.
If you look carefully after the balloon explodes, it appears that the wreckage falls on the OTHER side of the skydome. After Bond hits the dome, it is almost impossible to see, so look almost immediately after it explodes.