It's easy!

There are NO examples of a functional democracy surviving the kind of immigration trial you are describing, because NO functional democracy has a history that spans more than 300 years.

Your point about sharia courts in the UK is a sound one, but I would venture this observation: I have heard the Archbishop of Canterbury (Dr. Rowan Williams) advocate for a separate court in a half-hearted attempt to look tolerant and accommodating (much as he does any controversial topic in his own Church!). I have heard many fringe elements within British society call for a separate court system, claiming that the existing Courts (High and Low) are too biased by 1200 years of Christian affiliation. Of course, I have heard Muslim elements call for separate courts... for reasons too obvious to detail.

I have NOT heard even ONE representative voice of British Government voice serious consideration of this plan, though. No one from Downing Street, or Parliament, or the House of Lords seems to be taking the question seriously at all. They wouldn't do it for the Irish, the Scots, or the Welsh... I find it terribly difficult to believe that they would do it for a Muslim minority, no matter how vocal.