If you want to read about police firearms, the first half of the ACPO manual of guidance on the police use of firearms is on the ACPO website. The only bits that are subject to PII, and therefore not published, are those specifically relating to tactics. The rest, including ROE, are there for all to see.
Hope this answers a few questions. Ultimately though, the ultimate decision rests with the officer holding the gun. Strategic parameters are set at gold level, tactics, with the support of a tactical adviser, are set at silver, and the execution (unfortunate word) is carries out at bronze. At all levels, decisions are made on available intelligence at the time. If we take it that Gold strategy was to seek suicide bombers and stop atrocities, then silver had intelligence that house X contained persons believed to be bombers, and bronze officers on surveillance watched someone leave that house who went towards a railway station, and when challenged legged it on to a train, I think it is pretty easy to see how an officer could have a firmly held belief that another atrocity was about to take place.
Poor B*gger is about to spend the next twelve months in the washing machine. It is , a huge mess. I must say, I don't think this will be a SF operation either. This is doubtless a Met job.