GOLDEN RULE: Insurers should be informed of everything - then they can't wriggle.

If the vehicle is to be out of the home country for an extended period they should be told. Most UK insurance includes the very basic minimum legal cover (usually similar to RTA) in other EU countries at no additional cost for up to a certain number of days a year. Some insurers don't need to be told, others do. Most UK Certificates of Insurance have foreign wording on the reverse confirming this and should be accepted by other EU police forces.
Check your policy, even if going to Calais for an afternoons shopping. If you want equivalent cover abroad as you have at home they must be told, and they will no doubt charge something.

Not having a current MOT is not in itself sufficient reason for Insurers to decline a claim unless the claim arose from an incident that could have been influenced by mechanical soundness. eg. No MOT for ineffective lights generally would have no bearing on a daytime crash, neither does a lack of an MOT influence a thief - but Insurers will try it on !
An MOT can be taken by Insurers as a possible indicator of mechanical condition (and therefore value) even though the certificate states it is only met the legal minimum standard on the day of the test - which could have been 364 days ago.

Being on an SORN, in my view, would not affect insurance even if you are using it on UK roads - insurers only concern about your honesty is with them !

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