Part of the reason for the quick escalation is that (American) cops are taught things like the Tueller Drill https://www.youtube.com/watch?v=jwHYRBNc9r8 that claims that an attacker can close 21 feet in 1500 milliseconds and stab a victim so many times that even a fast ambulance response won't save him from bleeding out.
This idea means the cop has to unholster a pistol as soon as any sign of noncompliance is showm, start firing if a person "reaches for their waist", and empty the magazine because this Olympian attacker won't be stopped by a few bullets.
They just "want to get home to their families" despite the fact that car accidents are deadlier to cops and garbage men have more dangerous jobs.
Since UK police only exceptionally carry firearms, they have to play things safer. E.g. respond with more people. Keeping greater distance.
But lots of other police forces have - sometimes heavily - armed police with better results because they effectively act on the basis that using their weapon is an absolute last resort, and so you keep your distance if there's a risk they're armed, and call for backup rather than approach etc.