I think he is talking about immigration, not insurance. I wouldn't worry about it - just don't get picked up by the police for anything else. I'm pretty sure most countries don't take details of you leaving - you certainly don't show your passport when leaving the UK. And I can't see why they will care - you will be on the way out if anyone stops you.
Only one issue - if you do get stopped by the police for some other reason, you may not get another visa in the future. I doubt that trying an extension will make sense - if its even possible.
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