Actually, Canada does have double jeopardy laws - it's in Section 11 of the Charter. You can appeal a court decision (conviction or acquital) if you believe there was an error in judgment/application of law, but that is considered a continuation of the same trial. If an appeal is not granted, and new evidence later comes to light, you don't get a new trial. In the case of Joe Miller's trial, you could appeal on the grounds that excluding the confession was a legal error, and double jeopardy would only be an issue if the appeal is denied.
I don't know about the UK, but I'd assume it's similar since the justice systems have similar roots.
But yeah, it frustrated me to no end that they excluded the confession - such flimsy reasoning, and the prosecutor didn't even try to point that out.
Nigel and Beth are both irritating me.