The British basically concluded that ABM defences for the UK were nigh on impossible because of the physics of engaging the missiles. While they were initially based on IR and then later active radar systems to detect and engage the incoming BMs, they still needed to be launched, reverse direction and then attack the incoming missiles. This mean that engagements were more likely to occur over Western Europe than the UK or the North Sea and with the use of nuclear warheads, were politically unpalatable to the UK's NATO allies. So deterrence became the name of the game.
In the US's case, they had a large country and only one, friendly country between them and the launching country. So, they could afford to fire a missile and have it engage a ballistic missile head on, often with a nuclear warhead. However, even there, the physics have pretty much defeated their efforts to do that. Spartan and Sprint missiles had ridiculously short ranges, high accelerations and a nuclear warhead and they still weren't certain they could manage it. Now the ABM system in Alaska trades distance for engagement time (by engaging the missile in mid-flight). Even so, they have had to utilise various tricks to gain funding for it's development and test to make sure it actively engages the incoming ballistic missiles. Whether it would work in a real life situation is another matter.
