The police and the citizenry have massively different relationships in the UK and US. UK police appear to be far more trusted and liked by their citizens than American ones are.
Also, they appear to shoot their citizens less often, which probably helps.
Being a cop is a much more dangerous job than being a programmer, but it’s far from the most dangerous job around. Bartenders (1.6x), taxi drivers (1.7x), landscapers (1.5x), miners (2.6x), roofers (4x), fishers (7x), and loggers (9x) are all more likely to die on the job than cops, and none of these professions have developed the siege mentality that cops have.
Furthermore, the public image of patrol officer death typically involves violent shootouts with criminals, but in reality death by homicide only accounts for about half of patrol officer deaths, with death by automobile making up the rest. For 2013 10.8 officers died per 100,000, with 49% dying by car accident, 45% by homicide, and 6% by other causes.
This pales in comparison to the homicide rate for pregnant women, with up to 10.5 women being murdered per 100,000[0]. This means that a pregnant woman is as likely to be murdered as a cop is likely to die on the job from all causes, and said pregnant woman is twice as likely to be murdered than any cop.
I'm not sure that not being armed is that much different. Even if armed, if the principles behind the force are well set and you trust (based on facts and experiences) that that weapon will not leave the holster without need, it makes very little difference.
Just to give an example, I live in Serbia and policemen are armed. Police is far from clean, corruption is present as everywhere and there are obvious ties between crime and police. However, when it comes to the small guy, on the streets, on public events or encounters with an officer I can't say you should be cocerned with your own safety. Sometimes they are armed with rifles, when riskier events take place. Sometimes they are not the most pleasant. But I don't remember the last time I heard an officer discharge a weapon. Funny as it may seem, nobody wants dead bodies in his career. Pieceful de-escalation is a usual crisis outcome. There are a lot of veterans, PTSD sufferers and desperate people around here. Just think of the madness if police charges everywhere guns blazing... This went off topic...
Well the US citizens also carry guns a lot more often than the UK citizens.
So there’s your answer. But then constitutional rights yada yada. Just the idea that the other party had a deadly weapon, escalates things really fast.
Also, they appear to shoot their citizens less often, which probably helps.