While most U.S. Universities have strict deadlines for applying, most UK universities do not seem to mention any deadlines for applying on their websites. Why is this? Or am I mistaken? Before what date should I apply?

just apply, man. No deadline = apply asap

Most unis will accept applications ate any time of year. However, if you are after a funded PhD there will be deadlines. University and research council deadlines tend to be in the Feb-Apr period (for an October start) so make sure you know when they are and apply before.. However, some projects start in January so the deadlines will be in the autumn.

the sooner the better, if your what they want they will take you so dont hold off.