The British police developed in a very different environment than US policing. They take it as an article of faith (and historically it has worked) that most policing does not require firearms. Armed police are on call if a situation turns violent. (Much like SWAT backs up US police).
The issue is that in certain areas, crime is no longer the greatest problem. Instead, it is the threat of either armed attack on a high profile target or violent unarmed assaults by religious fanatics.
The Brits successfully coped with IRA terrorists, and still feel that they can handle the current threat with the same tactics. I hope they are correct, but I doubt it.
True, and it's impossible to overestimate the weight of history behind this. The culture is very deep-rooted in the Police, and anonymous polling of rank-and-file officers always shows a large majority against routine arming. That majority gets steadily smaller, however...
And don't forget how very few British police die as a result of violent crime - still only 26 since the year 2000.
Problem is they had acknowledged the IrA as a foreign like terror organization. They do not consider, however, terrorists who dream out loud of killing Britons as spies, so they let them spout it until they do it.