" When I am in one of your planes, even though I know I am only on my way to a court case, I feel like I should get up and ram the plane into a skyscraper. I can't help it. I'm a Muslim and that's how I think. "
Reading the article, it's not quite as bad as I initially thought. It sounds like this is just private arbitration, which happens to apply sharia. If it works in the UK anything like it works in the US, arbitration can't be used in a criminal case, and can only be used if both parties agree to resolve a dispute through arbitration.
Arbitration awards are enforceable under the principles of contract law: the parties agree to submit to arbitration, and abide by its ruling. If one party doesn't obey the ruling, the other party can take him to court, on a breach of contract theory.
Presumably, all of the "safety valves" that operate in contract law to prevent enforcement would also operate here: if the agreement to arbitrate was entered into under fraud, duress, or undue influence, it won't be enforced. Likewise, if the contract is for an illegal purpose, it won't be enforced (presumably if a sharia arbitrator sentences somebody to be stoned to death - which seems unlikely, given the types of cases that are being handled here - a British court won't enforce the judgment).
With all that said, I don't entirely approve of this. I fully support the right to engage in private arbitration, governed by whatever rules the parties agree to. If 2 parties voluntarily agree to have their dispute arbitrated under Sharia law, I support their right to do so. It affects nobody but them. But I'm pretty uncomfortable with a backwards system of law carrying any weight in a civilized nation, even in the roundabout way that arbitration judgments are given legal force, and even among those who "voluntarily" submit to it. Given the power imbalance that exists between men and women in the Muslim world, I'd question whether some of these agreements were voluntarily entered into.