OP is mistaken in thinking that native Anglophones wouldn't say I will fly to London after 10 days.
The only relevant factor here is that after [some amount of time] requires a context within which some particular point in time (past or future) has already been established. Thus...
I'll go to Glasgow on December the 12th, and stay with my aunt. I will fly to London after 10 days.
...is fine. The fact that this is a reference to future actions is irrelevant, since we can also say...
I went to Glasgow in June and stayed with my aunt. I flew to London after 10 days.
EDIT: Note that as it happens, I placed adverbial after BEFORE the relevant "duration, amount of time" (10 days, here), but syntactically speaking it would be fine (and mean exactly the same) if I'd placed it AFTER the duration (...fly / flew to London 10 days after. As with many adverbs in English, the exact position is flexible - particularly if there's no scope for alternative interpretations, as is the case here.
Also note that we can replace after by later without changing the meaning - but only as ...10 days later (the sequence later 10 days is never valid in English).
EDIT2: IMHO, the cited usage in the BBC article is poorly phrased, and even many native Anglophones might struggle to answer "10 days after what?" (in the context of "When will the policy of keeping schools open be reassessed?").
Logically, the only interpretation that makes sense is The policy of keeping schools open will be reassessed 10 days after the lockdown is introduced on Monday 22nd Nov (i.e. - the reassessment is scheduled to take place on 2nd Dec). But the way it's phrased strongly steers us to assume the reassessment will take place ten days after the most recently-mentioned date (12th Dec), even though logically, that makes no sense.
Syntactically, it's even possible that what's to be reassessed after 10 days is the entire lockdown policy - but that's just another reason why I say the text isn't well constructed. I'm as certain as I can be that the intention is to say that specifically the policy in relation to schools should be reassessed after 10 days.