When you get arrested in England, you do have a right to remain silent, but it might be used against you. The police will typically say:
You do not have to say anything, but it may harm your defence if you do not mention when questioned something which you later rely on in court.
I've never really thought of it as a bad thing. You can still say nothing, or "I want to speak with a lawyer before I answer any questions".
I've always considered it to be more about a line of questioning at trial. If you've invented an alibi, the prosecution can ask you "why are you only coming up with this excuse now but you didn't at the time?". You might then have a good reason for this, such as "I didn't think it was a good idea to answer questions without first consulting a lawyer".
The American model of having immunity if you just stay quiet is a bit odd in comparison.