If the journal is already in SCI/SCIE indexed then there is no point to include/exclude Web of Science list. SCI/SCIE is one of the top indexes to check the class of journal. This is what I understand.
Once the issue got published in print (FIN) it will reach Web of Science (WoS) within 4-8 weeks. Web of Science adds them in batches. All the journals with Impact Factor (IF) should transfer their article to web of science. Also it is not a mandatory condition that journals without IF will not share papers to WoS, journals can share articles to WoS even if they don’t have IF.