If you listen to the video, the professor says that nothing you say to the US police can be used to defend you later.

From your quote we see that this is not the case in the UK. Something you say can later be used in your defense, so clearly if you choose to say nothing -- you have less that could be used in your defense.

> I imagine there is a sizable amount of injustice in the UK due to not having a 4th amendment like rights.

Compared to the US?

No, we have hearsay laws similar to the US. We can't compel an officer to testify on our behalf about something we said to them in questioning. its just in the UK, if you talk to the police, and forget something that is important you don't get to use it later, or its not as useful later.

This has lead to a lot of unsound convictions of people who were intoxicated at the time of arrest; They tell some abridged version of their story to police, leaving half the important details out, leaving them with half a case to take to court. In Scotland they have started Requiring solicitors to be present at interviews of suspects (of some crimes) because of this, making statements inadmissible unless a solicitor was present.