They are continuing the same employment, so they should just ensure their payments continue. If they take a lump sum, they would have to start all over again.
To be belt and braces, get a letter from your employer stating your last day of employment as ... Oct , then cancel your work permit, then go and cancel your extension, this can be done in advance,
OK you are on an extension and not a visa. There is no short term extension. If you want to apply in London for a new Non-B then you need to cancel this extension and your work permit, and have all the paperwork including a WP3 from your employer. I really don't think you have any options really, (let's hope I am wrong).