Now that Thameslink no longer goes via London Bridge, you are relying being able to make your own way across central London on a NOT UNDERGROUND ticket. This is a bit of a grey area, and there have been a few threads on it in the past.
Here is one such thread
I have a NOT UNDERGROUND season ticket used every day, and I've only been chtallenged once at London Bridge with it, but this was when Thameslink still went via London Bridge, and another member of staff backed me up.
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Actually, I'm going to retract that. The route is presumably still mapped in the Routeing Guide, so is there anything that says a mapped route becomes invalid if part of the mapped route has no scheduled passenger services any more? I believe not, so it would be valid without having to get into the crossing argument above