http://en.wikipedia.org/wiki/Queen's_Flight
It is operated by the RAF
Wikipedia indicates this, however:
International and intercontinental travel had traditionally been made using a Vickers VC-10 airliner from the squadron; however as those aircraft have grown older and obsolete, with many now out of service, the Prime Minister has increasingly turned to chartering aircraft from British Airways (most commonly a Boeing 777 is chartered).
It also says that Blair proposed a dedicated Airbus for his and the Queen's use; however, the money will not be appropriated until at least 2008.
So, it appears Blair does use chartered BA jumbos.