I can't see EC/261 applying here since the operating airline that was late here was AA and they are outwith the Regulations in this instance. One exception to this is that the courts have held that where an airline rebooks passengers to a later flight and yet the passenger could have got to the gate in time, then that can be regarded as Involuntary Denied Boarding. If AA did that rather than BA, it's still not going anywhere, and in any case it appears BA did accept those who got to the gate.
The bus service is definitely Qantas only, I believe that is due to a cost sharing agreement between AA and Qantas, so if BA passengers started using it I can see there would be an issue there. More to the point I suspect that the AA agents at gate 44 may not have been fully informed - their main job is to stop BA passengers from getting on that bus! This, incidentally, is a temporary issue, in December 2015 / January 2016 a new airside tunnel / walkway is scheduled to open up between T4 and TBIT for all passengers.
For HBO passengers who were quick on their feet, and assuming TBIT's security was quick, then it would have been possible (just) to go the usual landside route. Getting out of T4 can be done in just a few minutes, the walk to TBIT under 5 minutes. TSA, well that's a variable, but 10 minutes seems right to me, plus 10 minutes to the BA gate. So 30 minutes covers it, BA would want people at the gate no later than 15 minutes before departure, so we are up to 45 minutes.
So add in checked luggage to the mix, some people a bit slow on their feet, most of them unfamiliar with LAX's layout and routes, then when the decision was made, AA arrival was 20:50, BA departure 21:30, different terminals, that's not feasible. The 15 minute delay - which I guess was apparent late in this sequence - was a bit of game changer for some, but as we know, many passengers get very irked if their baggage is delayed. Or indeed if they have been tipped out of their favourite seat!
So in summary, this was a bit unfortunate, as is often the case, but I don't blame BA (or AA) in deciding the connection was infeasible for most. For some it would have been feasible as we saw, and they were accommodated, but in way it was self selecting: those who didn't/couldn't rush were sorted out very efficiently, those that got through were allowed on their flight.
How could it have been handled better? Ideally a special bus could have been laid on from the rear steps of the AA service then going to TBIT, but that breaks all the local rules in LAX, it's not going to happen. Whoever suggested gate 44 may not have had the power to fix the contract issues with Qantas on that one (and it doesn't run that frequently either).
So until the tunnel opens, maybe it would have been best to announce that everyone had been rebooked and to collect their hotel vouchers. Once airborne at LAS the infeasible timetable would have been clear. But then they could add that anyone with boarding passes, HBO/not bothered about luggage, was welcome to try going landside but the chances are they would not make it. That strikes me as more transparent.