How's this for a possible scenario :
They start following this bloke because he's come out of an address which they have intelligence is dodgy. He's not a specific target, but he's wearing a warm jacket in summer and perhaps fits somebody's profile of a terrorist (dark skin etc.) so they want to keep an eye on him. They follow him on his bus ride but when he enters the tube station they decide enough is enough and challenge him. At this point he freaks and legs it towards the nearest train, jumping a ticket barrier in the process, and the police sh1t themselves as it now looks like they have a desperate suicide bomber on their hands determined to do maximum damage. So they chase him and shoot him before he manages to get into a carriage.
This is obviously pure speculation, but in running like he did he maybe went from possible to definite terrorist very quickly in the eyes of the police and gave them very little time in which to make a decision. I still think they made the right decision given the circumstances, but its a terrible price to pay for overstaying your visa.
There's also not much of an 'intelligence failure', a favourite of the media, in this scenario as he was wasn't a definite target until he scarpered.