Those three horizontal lines are definitely not something you see every day; they're far, far too shallow and narrow to have anything to do with the driving bands. I have an 8,8cm Sprgr. 4,5 Stg shell on my desk and it has nothing like that, so I can rule that one out for you. I've also looked at all the images I have in my database and there's not one German shell from 5cm to 12,8cm that has any such feature- even the controlled-fragmentataion 8,8cm shell has only one horizontal groove. So if this is a shell fragment and not a bomb fragment, it doesn't seem likely to be German.
If kept by a member of the RAF, they could well be British as it'd be their AA guns defending that'd put a lot of fragments on the ground- far more easily collected than German pieces that'd have to be stuck in bombers that made it home (although maybe there wasn't any shortage of those either). But I'd suggest a British ordnance expert might be able to work with the lines as they are an identifying feature to be sure.
Oh and what exactly is the wall thickness? The photos with the rule are certainly good, but since neither of the fragments is specifically on edge, it's hard to tell. Having been deformed by its detonation has surely affected the size somewhat, but perhaps it'll help rule something out at least.
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