Notice, that with this meaning, the word is an uncountable noun, so you can only say youth, not youths. (BBC)
Does "that" indicate that the whole part afterwards is the objective clause of the verb "notice"? But why there is a comma between "that" and "notice"?
Correct: only "that with this meaning, the word is an uncountable noun" is the object/complement of "notice".
It's difficult to challenge the BBC but I would certainly not put a comma there to avoid misunderstanding.
Notice what? Yes, you got it right! Here, 'that' indicates the rest of the part to pay attention to.
A similar pattern is the result from COCAE:
Notice that with equal-weighting, the sum of the attribute values in Table I would be 0 for each system and, therefore,...
Notice, [that with this meaning, the word is an uncountable noun, so you can only say youth, not youths].
The comma after "notice" is not required.
This is an imperative construction where "notice" is the verb and the bracketed element is a subordinate content clause functioning as complement of "notice".
Traditional grammar treats the that clause as direct object of "notice", but modern grammar takes it as a complement.
With only a few minor exceptions, direct objects are always noun phrases, not clauses.
