You seem to imply that cops in the US _are_ allowed to lie, and search your private property without cause.

If a cop in the US lies to you, or searches your car without cause, or otherwise 'tricks' you, it's no more admissible that it would be in the UK. The issue (in both countries) is proving the cop lied to you, tricked you, etc. Which is the whole point of have legal consul present in the first place.

One of the comments on the article mentions that the 'UK version of the "Miranda rights" includes the sentence "It may harm your defense if you do not mention, when questioned, something which you later rely on in court."' The most common tactic I've heard cops employ in the US is "if you're innocent, it's in your best interest to talk to us." It would appear the UK has that trick built right in to the process from the start.